m 


2   - 


<    u 


tn 


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J-      CM 

o  2 


L 


LJ 


H. 


TF 


34- 


PBEFACE. 


ALTHOUGH  the  modern  railway  system  is  but  about  fifty 
years  old,  yet  its  growth  has  been  so  rapid,  and  the  progress 
in  the  science  of  railway  construction  so  great,  as  to  render  the 
earlier  technical  books  on  this  subject  inadequate  to  the  needs 
of  the  engineer  of  to-day. 

In  the  course  of  his  practical  experience  as  a  railway  engi- 
neer, the  author  was  strongly  impressed  with  the  want  of  a 
more  complete  hand-book  for  field  use,  and  finally  concluded, 
at  the  solicitation  of  his  friends,  to  undertake  the  preparation 
of  the  present  volume. 

The  aim  in  this  work  has  been: 

First — To  present  the  general  subject  of  railway  field  work 
in  a  progressive  and  logical  order,  for  the  benefit  of  beginners. 

Second — To  classify  the  various  problems  presented,  so  that 
they  may  be  readily  referred  to. 

Third — To  embrace  discussions  of  all  the  more  important 
practical  questions  while  avoiding  matters  non-essential. 

Fourth — To  employ  throughout  the  work  a  uniform  and 
systematic  notation,  easily  understood  and  remembered,  so 
that  after  one  perusal  the  formula  may  be  intelligible  at  a 
glance  wherever  referred  to. 

Fifth— To  express  the  resulting  formula  of  every  problem 
in  the  shape  best  adapted  to  convenient  numerical  compu- 
tation. 

Sixth — To  furnish  a  large  variety  of  useful  tables,  more  com- 
plete and  extended  than  any  heretofore  published,  especially 
adapted  to  the  wants  of  the  field  engineer. 

An  elementary  knowledge  of  algebra,  geometry  and  trigono- 
metry on  the  part  of  the  reader  has  been  taken  for  granted,  as 
a  command  of  these  instrumentalities  is  deemed  essential  to 
the  education  of  the  civil  engineer.  The  few  references  to 
mechanics,  analytical  geometry,  optics  and  the  calculus  may 
be  assumed  correct  by  those  not  conversant  with  these 
branches. 

0  1823 


•** 

IV  PREFACE. 

Many  of  the  problems  in  curves  are  new,  yet  there  is  hardly 
one  that  has  not  presented  itself  to  the  author  in  the  course  of 
his  practice.  The  investigation  of  the  valvoid  curve  is  original, 
and  though  the  mathematical  discussion  is  somewhat  difficult, 
yet  the  resulting  formulae,  taken  in  connection  with  Table  X, 
are  exceedingly  simple  and  convenient  for  the  solution  of  a 
certain  class  of  problems. 

The  treatment  of  compound  curves  is  novel  and  exhaustive. 
A  few  general  equations  are  established,  which,  by  slight 
modifications,  solve  all  the  problems  that  can  occur. 

No  discussion  of  reversed  curves  is  given,  because  these  .are 
inconsistent  with  good  practice,  except  in  turnouts,  under 
which  head  they  are  noticed. 

The  chapter  on  levelling  includes  a  discussion  of  stadia 
measurements,  with  practical  formula.  The  chapter  on  earth- 
work contains  a  review  of  several  methods  for  calculating 
quantities,  and  states  the  conditions  under  which  these  suc- 
ceed or  fail  in  giving  correct  results. 

Among  the  tables,  numbers  3,  5,  6,  10,  18,  19,  26  and  29 
are  original.  The  adoption  of  versed  sines  and  external 
secants  throughout  the  work,  wherever  these  would  simplify 
the  formulae,  rendered  necessary  the  preparation  of  tables  of 
these  functions.  The  table  of  logarithmic  versed  sines  and 
external  secants  has  been  computed  from  ten-place  logarithmic 
tables  of  sines  and  tangents,  so  that  the  last  decimal  is  to  be 
relied  on,  and  no  pains  have  been  spared  to  make  the  table 
thoroughly  accurate. 

Tables  numbers  4,  7,  8,  9,  11,  12,  13,  14  and  30  have  been 
recalculated,  enlarged,  and  some  of  them  carried  to  more  deci- 
mal places  than  similar  tables  heretofore  published.  The 
intention  has  been  to  give  one  more  decimal  than  usual,  so  that 
in  any  combination  of  figures  the  result  of  calculation  might 
be  reliable  to  the  last  figure  usually  required. 

The  tables  which  have  been  compiled  and  rearranged  are 
numbers  1,  2,  15,  16,  17,  24,  25  and  31.  The  tables  of  log. 
sines  and  tangents  here  given  are  the  only  six-place  tables 
which  give  the  differences  correctly  for  seconds.  The  table 
of  logarithms  of  numbers  is  accompanied  by  a  complete  table 
of  proportional  parts,  which  greatly  facilitates  interpolation 
for  the  fifth  and  sixth  figures. 

In  all  the  tables,  whether  new  or  old,  scrupulous  care  has 


PREFACE.  V 

been  taken  to  make  the  last  figure  correct,  and  the  greatest 
diligence  lias  been  exercised  by  various  checks  and  compari- 
sons to  eliminate  every  error.  It  is,  therefore,  hoped  and 
believed  that  a  very  high  degree  of  accuracy  has  been  ob- 
tained, and  that  these  tables  will  be  found  to  stand  second  to 
none  in  this  respect. 

The  preparation  of  this  work  has  extended  over  several 
years,  as  time  could  be  spared  to  it  from  other  engagements. 
It  is,  therefore,  the  expression  of  deliberate  thought,  based  on 
experience,  and  as  such  is  submitted  to  the  judgment  of 
brother  engineers.  If  it  shall  prove  to  have  even  partially 
met  the  aim  herein  announced,  and  so  shall  serve  to  smooth 
the  way  of  the  ambitious  student,  or  to  assist  the  expert  in  his 
responsible  duties,  the  labors  of  the  author  will  not  have  been 
in  vain.  WM.  IL  SEARLES,  C.E. 

NEW  YORK,  March  1st,  1880. 


TABLES. 


I.  Geometrical  Propositions 271 

II.  Trigonometrical  Formulae 273 

HI.  Curve  Formulae 277 

IV.  Radii,  Offsets,  and  Ordinates 280 

V.  Corrections  for  Tangents  and  Externals 288 

VI.  Tangents  and  Externals  to  a  One-Degree  Curve 289 

VTI.  Long  Chords  and  Actual  Arcs 293 

VIII.  Middle  Ordinates  to  Long  Chords 298 

IX.  Linear  Deflections •  •  301 

X.  Curved  Deflections;  Valvoid  Arcs 302 

XI.  Frog  Angles  and  Switches 393 

XII.  Middle  Ordinates  for  Rails 304 

XIII.  Difference  in  Elevation  of  Rails 304 

XIV.  Grades  and  Grade  Angles .". 305 

XV.  Barometric  Heights,  in  feet 307 

XVI.  Coefficient  of  Correction  for  Atmospheric  Temperature....  309 

XVII.  Correction  for  Earth's  Curvature  and  Refraction 309 

XVIII.  Coefficient  for  Reducing  Stadia  Measurements 310 

XIX.  Logarithmic  Coefficients,  Stadia  Measurements 311 

XX.  Lengths  of  Circular  Arcs 312 

XXI.  Minutes  in  Decimals  of  a  Degree 313 

XXII.  Inches  in  Decimals  of  a  Foot 314 

XXIII.  Squares,  Cubes,  Roots  and  Reciprocals 315 

XXIV.  Logarithms  of  Numbers 333 

XXV.  Logarithmic  Sines,  Cosines,  Tangents,  and  Cotangents. 359 

XXVI.  Logarithmic  Versed  Sines,  and  External  Secants, 404 

XXVII.  Natural  Sines  and  Cosines 449 

XXVHI.  Natural  Tangents  and  Cotangents 458 

XXIX.  Natural  Versed  Sines  and  External  Secants 470 

XXX.  Cubic  Yards  per  100  Feet  in  Level  Prismoids 493 

Useful  Numbers  and  Formulae 498 


CONTENTS. 


CHAPTER  I. 

RECONNOISSANCE. 

SECTION  PAGE 

2.  Topographical  considerations 1 

6.  Use  of  maps 3 

7.  Pocket  instruments 3 

9.  Aneroids 4 

10.  Formulae  for  aneroid 4 

16.  Locke  level 7 

17.  Prismatic  compass 7 


CEJATTER  II. 

PRELIMINARY  SURVEY. 

18.  Definitions 8 

19.  Engineer  corps 8 

20.  Chief  engineer 8 

21.  Assistant  engineer 9 

22.  Chainman 10 

24.  Axeman 11 

26.  Topographer 11 

28.  Leveller 13 

29.  Rodmen 13 

32.  Thecompass 14 

33.  The  chain 14 

35.  The  level 15 

36.  Therods 15 

38.  The  clinometer 16 

39.  The  plane-table  16 

40.  The  transit  16 

42.  Transit  points 17 

43.  Transit  flags 17 

44.  Obstacles  to  alignment  and  measurement : 18 

45.  Parallel  lines 18 

46.  Lines  at  a  small  angle - . . . .  18 

47.  General  problem 19 

48.  Lines  at  a  large  angle 20 

49.  Selection  of  angles 21 

53.  Rocky  shores;  Tie-lines 22 

54.  System  of  plotting  map 23 


Vlll  CONTENTS. 

CHAPTER  III. 

THEORY  OF  MAXIMUM  ECONOMY  IN  GRADES  AND  CURVES. 
SECTION  PAGE 

55.  Choice  of  routes 24 

56.  Statement  of  the  problem 24 

57.  Engine  traction 25 

58.  Engine  expense 26 

60.  Resistance  to  motion 27 

61.  Resistance  due  to  grade 27 

62.  Resistance  due  to  curve  28 

63.  Formulae  for  resistance 28 

64.  Formulae  for  maximum  trains 21? 

65.  Engine-stage 30 

66.  Graphical  solution 32 

67.  Train  load  reciprocals 33 

68.  Reduction  of  grades  on  curves 33 

69.  Example 33 

70.  Assisting  engines 36 

71.  Maximum  return  grades 36 

72.  Undulating  grades 37 

75.  Comparison  of  routes :~ 38 

76.  Value  of  distance  saved 39 

77.  Conclusion 39 

CHAPTER  IV. 

LOCATION. 

78.  General  remarks 39 

79.  Long  tangents 39 

80.  Leveller's  duties;  Profiling 40 

81.  Establishing  grade  lines 41 

CHAPTER  V. 

SIMPLE   CURVES. 

A.  Elementary  Relations, 

82.  Limits  to  curves  and  tangents 42 


83.  Definition  of  terms 42 

84.  Radius  and  degree  of  curve 43 

65.  Measurement  of  curves 44 

8G.  Approximate  value  of  R 44 

87.  Central  angle  and  length  of  curve 45 

88.  Definition  of  other  elements 4* 

90.  Formula  for  tangent  distance  T 46 

91.  Formula  for  long  chord  C —  .- 47 

92.  Formula  for  middle  ordinate  M 48 

93.  Formula  for  external  distance  E 48 

95.  Formula  for  radius  in  terms  of  Tand  A 49 

96.  Formula  for  external  distance  in  terms  of  T  and  A 50 

97.  Formula  for  radius  in  terms  of  E  and  A 50 


CONTENTS. 


SECTION  PAGE 

98.  Formula  for  tangent  distance  in  terms  of  E  and  A  .......  ......  51 

99.  To  define  the  curve  of  an  old  track  ...............................  51 

100.  Other  curve  formula;  Table  III  ..................................  52 

B.  Location  of  Curves  by  Deflection  Angles. 

101.  Deflection  angles  .................................................  52 

102.  Rule  for  deflections  .....................  ,  ........................  53 

103.  Rule  for  finding  direction  of  tangent  at  any  point  ..............  53 

104.  Subchords  .........................................................  54 

105.  Deflections  for  subchords  .........................................  54 

106.  Correction  for  subchords  .........................................  55 

107.  Ratio  of  correction  to  excess  of  arc  ..............................  55 

108.  Transit  work  on  curves  ...........................................  57 

109.  Field  notes  ........................................................  58 

110.  Central  angle  in  terms  of  deflections  .............................  58 

111.  Method  by  deflections  only  ......................................  58 

C.  Location  of  Curves  by  Offsets. 

112.  Fourmethods  ....................................................  59 

113.  By  offsets  from  the  chords  produced  .............................  59 

114.  Do.  beginning  with  a  subchord  ...................................  60 

115.  Formula  for  subchord  offsets,  approximate  .....................  61 

117.  By  middle  ordinates  ..............................................  61 

1  18.  Do.  beginning  with  a  subchord  ...................................  G2 

119.  By  tangent  offsets  ........................................  .  .......  62 

120.  Do.  beginning  with  a  subchord  ...................................  64 

121.  By  ordinates  from  a  long  chord  ..................................  64 

122.  Do.  for  an  even  number  of  stations  ...............................  65 

123.  Do.  for  an  odd  number  of  stations  ................................  66 

124.  Do.  for  an  even  number  of  half  stations  .........................  67 

125.  Do.  beginning  with  any  subchord  ................................  67 

126.  Erecting  perpendiculars  without  instrument  .....................  69 

D.  Obstacles  to  the  Location  of  Curves. 

127.  The  vertex  inaccessible  ...........................................  69 

128.  The  point  of  curve  inaccessible  ...................................  70 

129.  The  vertex  and  point  of  curve  inaccessible  ......................  70 

130.  The  point  of  tangent  inaccessible  .........................  •  .......  71 

131.  To  pass  an  obstacle  on  a  curve  ...................................  72 

E.  Special  Problems  in  Simple  Curves. 

132.  To  find  the  change  in  R  and  E  for  a  given  change  in  'T.  ..........  73 

133.  To  find  the  change  in  R  and  T  for  a  given  change  in  E.  ..........  74 

134.  To  find  the  change  in  T  and  E  for  a  given  change  in  R  ...........  75 

135.  General  expression  for  elementary  ratios  ........................  75 

136.  To  find  a  new  point  of  curve  for  a  parallel  tangent  ..............  76 

137.  To  find  a  new  radius  for  a  parallel  tangent  ......................  76 

138.  To  find  new  P.  C.  and  new  radius  for  a  parallel  tangent  ..........  77 


X  COKTEKTS. 

SECTION  PAGE 

139.  To  find  new  tangent  points  for  two  parallel  tangents 78 

140.  To  find  new  R  and  P.  C,  for  new  tangent  at  same  P.  T. 80 

141.  To  find  new  P.  C.  for  a  new  tangent  from  same  vertex 81 

142.  To  find  new  radius  for  a  new  tangent  from  same  vertex 81 

143.  To  find  new  R  and  P.O.  for  same  external  distance,  but  new  A .  82 

144.  To  find  a  curve  to  pass  through  a  given  point 83 

145.  To  find  new  radius  for  a  given  radial  offset 84 

146.  Equation  of  the  valvoid- 86 

147.  To  find  direction  of  a  tangent  to  the  valvoid  at  any  point 87 

148.  To  find  the  radius  of  curvature  of  the  valvoid  at  any  point 88 

149.  To  find  the  length  of  arc  of  the  valvoid 88 

150.  To  find  new  position  of  any  stake  for  a  new  radius  from  same 

P.C 89 

151.  To  find  new  radius  from  same  P.  0.  for  new  position  of  any 

station 92 

152.  To  find  distance  on  any  line  between  tangent  and  curve 93 

153.  To  find  a  tangent  to  pass  through  a  distant  point 94 

154.  To  find  a  line  tangent  to  two  curves 96 

155.  To  find  a  line  tangent  to  two  curves  reversed 98 

156.  Study  of  location  on  preliminary  map ;  Templets ;  Table  of  con- 

venient curves ". 100 


CHAPTER  VI. 

COMPOUND  CURVES. 

A.  Theory  of  Compound  Curves. 

157.  Definition 102 

158.  The  circumscribing  circle 102 

159.  The  locus  of  the  point  of  compound  curve : 103 

160.  The  inscribed  circle  of  the  principal  radii 104 

Cor.  2.  Maxima  and  minima  of  the  radii  104 

B.  General  Equations. 

161.  Formula  for  radii,  central  angles,  and  sides 105 

162.  Given:  Si  Sa  A  and  R^  to  find  A  x   A  2  and  Rz 106 

163.  Given:  AB,  VAB,  VBA   and  R3,  to  find  A  2    Ala.ndRl 107 

164.  Given:  7?t  A  ^  R2  A 2,  to  find  the  triangle  VAB 108 

165.  Given:   A ,  the  radii,  and  one  side  to  find  the  other. . .   108 

166.  Given:'  one  side,  radius  and  central  angle  to  find  the  others 110 

167.  Remarks  on  special  cases Ill 

168.  Obstacles;  theP.C.C.  inaccessible 112 

C.  Special  Problems  in  Compound  Curves. 

169.  To  find  a  new  P.C.C.  for  a  parallel  tangent 113 

170.  To  find  a  new  P.C.C.  and  last  radius  for  a  parallel  tangent 115 

171.  To  find  a  new  P.C.C.  and  last  radius  for  the  same  tangent 118 

172.  To  find  a  new  P.C.C.  and  last  radius  Ra'  for  new  direction  of 

tangent  through  same  P.T 121 


COKTEKTS.  XI 


SECTION  PAGE 

173.  To  find  a  new  P.O. C.  and  Jast  radius  -R/  for'new  direction  of 

tangent  through  same  P.T 124 

174.  To  replace  a  simple  curve  by  a  three-centred  compound  curve 

between  the  same  tangent  points 127 

175.  To  find  the  distance  between  the  middle  points  of  a  simple  curve 

and  three-centred  compound  curve 129 

176.  To  replace  a  simple  curve  by  a  three-centred  compound  curve 

passing  through  the  same  middle  point 129 

I.  The  curve  flattened  at  the  tangents 129 

II.  The  curve  sharpened  at  the  tangents 132 

177.  To  replace  a  tangent  by  a  curve  compounded  with  the  adjacent 

curves 134 

I.  When  the  perpendicular  offset  p  is  assumed 136 

II.  When  the  angle  ct  or  ft  is  assumed 137 

III.  When  the  radius  Rz  is  assumed 137 

IV.  Locus  of  the  centre  O2 138 

178.  To  replace  the  middle  arc  of  a  three-centred  compound  by  an 

arc  of  different  radius 140 

I.  When  the  radius  of  the  middle  arc  is  the  greatest 140 

II.  When  the  radius  of  the  middle  arc  is  the  least 141 

III.  When  the  radius  of  the  middle  arc  is  intermediate 142 


CHAPTER  VII. 
TURNOUTS. 

179.  Definitions;  Frogs  and  switches 147 

180.  Single  tuwiout  from  straight  track  in  terms  of  frog  angle  148 

181.  Single  turnout  from  straight  track  in  terms  of  frog  number 149 

182.  Double  turnout,  middle  track  straight,  to  calculate  F" 151 

183.  Double  turnout,  middle  track  straight,  and  three  given  frogs. . .  152 

184.  Double  turnout  on  same  side  of  straight  track  to  calculate  the 

middle  frog,  F" 153 

185.  Double  turnout  on  same  side  of  straight  track  with  three  given 

frogs 155 

a.  When  the  middle  track  is  a  simple  curve 155 

6.  When  the  middle  track  is  straight  beyond  F 158 

c.  When  the  middle  track  is  reversed  at  F. 159 

186.  Turnout  on  the  inside  of  a  curved  track 101 

187.  Turnout  on  the  outside  of  a  curved  track 1G3 

188.  Tongue  switches 164 

189.  Tongue  switch  turnout  from  a  straight  track '. 164 

190.  Tongue  switch  double  turnout  to  find  F" 165 

191.  Tongue  switch  double  turnout  with  three  given  frogs 166 

192.  Tongue  switch  double  turnout  on  same  side  of  straight  track 

with  three  given  frogs 167 

a.  The  middle  track  reversed  at  F 167 

6.  The  middle  track  compounded  at  F 168 

c.  The  middle  track  straight  beyond  F 168 


Xll  CONTENTS. 

SECTION  PAGE 

193.  To  find  the  reversed  curve  for  parallel  siding  in  terms  of  .Fand 

perpendicular  distance  p 169 

194.  To  find  the  connecting  curve  from  frog  to  parallel  siding  on  a 

curve  in  terms  of  F  and  perpendicular  distance  p 170 

a.  The  siding  outside  of  main  track 171 

b.  The  siding  inside  of  main  track 171 

195.  To  locate  a  crossing  between  parallel  tracks 172 

196.  To  locate  a  reversed  curve  crossing  between  straight  tracks 173 

197.  To  locate  a  reversed  curve  crossing  between  curved  tracks 174 

198.  To  find  the  middle  ordinate  m,  for  one  station  in  terms  of  D 175 

199.  To  find  the  middle  ordinate  mi  for  rails,  in  terms  of  rail  and  R..  175 

200.  Curving  rails ;  To  find  mi  in  terms  of  rail  and  m 176 

201.  To  find  elevation  of  outer  rail  on  curves 177 

202.  To  find  a  chord  whose  middle  ordinate  equals  the  proper  eleva- 

tion    179 

203.  General  remarks  on  elevation  of  rail 179 

204.  General  remarks  oa  coned  wheels 180 

CHAPTER  VIIL 
LEVELLING. 

205.  Use  of  the  engineers'  level 181 

206.  The  datum,  how  assumed 181 

207.  Benches,  how  used;  B.M 181 

208.  Height  of  instrument;  H.I 182 

£09.  Reading  of  the  rod 182 

210.  Elevation  of  intermediate  points 182 

211.  Turning  points ;  T.P. 182 

212.  Rule  for  backsights  and  foresights .* 183 

213.  Form  of  field-book;  proof  of  extensions 183 

214.  Profiles 184 

215.  Simple  levelling;  test  levels 185 

216.  Errors  in  reading,  'due  to  the  level ;  how  avoided 185 

217.  Errors  in  reading,  due  to  the  rod ;  how  avoided 185 

218.  Errors  due  to  curvature  of  the  earth 1 86 

219.  Errors  due  to  refraction 187 

220.  Radius  of  curvature  of  the  earth 187 

221.  Levelling  by  transit  or  theodolite 188 

222.  To  find  the  H.  I.  by  observation  of  the  horizon 189 

223.  Stadia  measurements ;  horizontal  sights 191 

224.  Stadia  measurements ;  inclined  sights,  vertical  rod 193 

225.  Stadia  measurements;  inclined  sights,  inclined  rod 1&5 

CHAPTER  IX. 
CONSTRUCTION. 

228.  Organization  of  engineer  department 196 

227.  Clearing  and  grubbing 197 

228.  Test  levels  and  guard  plugs , 197 


CONTEXTS.  xiii 


SECTION  PAGE 

2^9.  Cross  sections;  Slopes 197 

230.  Cross  sections,  formulae  for 198 

231.  Cross  sections,  staking  out 200 

232.  Cross  sections  on  irregular  ground 201 

233.  Cross  sections  on  side-hill  work 201 

234.  Compound  cross  sections , 202 

235.  Selection  of  points  for  cross  sections 203 

236.  Vertical  curves 203 

237.  Form  of  cross-section  book 204 

238.  Extended  cross  profiles 205 

239.  Inaccessible  sections 205 

240.  Isolated  masses 206 

211.  Borrow-pits 206 

242.  Shrinkage ;  Increase 206 

243.  Office-work 207 

244."  Alteration  of  line 207 

245.  Drains  and  culverts 208 

246.  Arch  culverts 209 

247.  Foundation  pits;  Bridge  chords  on  curves 210 

248.  Cattle-guards 214 

249.  Trestle-work 214 

250.  Tunnels:    Location;    Alignment;    Shafts;     Curves  ;     Levels  ; 

Grades;  Sections;   Rate  of  progress;  Ventilation;  Drain- 
age    216 

251.  Retracing  the  line 222 

252.  Side  ditches  and  drains 223 

253.  Ballasting 223 

254.  Track-laying ;  Expansion  of  rails ;  Sidings 223 

CHAPTER  X. 
CALCULATION  OF  EARTHWORK. 

254.  Prismoids ;  Choice  of  cross  sections 225 

255.  Formulae  for  sectional  areas 227 

256.  Prismoidal  formulae  for  solid  contents 229 

257.  Tables  of  quantities  in  cubic  yards 229 

258.  Tables  of  equivalent  depths 231 

259.  Formula  for  equivalent  depth  in  terms  of  slope  angle '. 232 

260.  Conditions  necessary  for  correct  results  in  use  of  tables 233 

261.  Method  of  mean  areas ;  correction  required 233 

262.  Exact  calculation  of  content;  examples 234 

2C3.  Wedges  and  pyramids 236 

264.  Side-hill  sections,  uniform  slope '. 236 

265.  Side-hill  sections,  irregular  ground 237 

206.  Side-hill  sections  in  terms  of  slope  angle 237 

207.  Systems  of  diagrams 238 

268.  Correction  for  curvature  in  earthwork 239 

269.  Haul ;  Centre  of  gravity  of  prismoid 243 

270.  Final  estimate , 245 

271.  Monthly  estimates. . , 246 


XIV  COHTEKTS. 

CHAPTER  XI. 

TOPOGRAPHICAL  SKETCHING. 
SECTION  PAGE 

272.  General  remarks 247 

273.  Artificial  features 248 

274.  Natural  features ;  Contours ;  Hatchings 248 

275.  Method  of  sketching 249 

CHAPTER  XH. 
ADJUSTMENT  OF  INSTRUMENTS. 

276.  The  transit 250 

277.  The  level 252 

278.  The  theodolite 253 

CHAPTER  XIII. 
EXPLANATION  OF  THE  TABLES 253 


FIELD 


CHAPTER  I. 
RECONNOISSANCE. 

1.  The  engineering  operations  requisite  to  and  preceding  the 
construction  of  a  railroad  are  in  general : 

THE  RECONNOISSANCE, 

THE  PRELIMINARY  SURVEY,  and 

THE  LOCATION.  „ 

2.  The  Reconnaissance   is  a  general  and  somewhat  hasty 
examination  of  the  country  through  which  the  proposed  road 
is  to  pass,  for  the  purpose  of  noting  its  more  prominent 
features,  and  acquiring  a  general  knowledge  of  its  topography 
with  reference  to  the  selection  of    a  suitable  route.     The 
judicious  selection  of  a  route  may  be  a  very  simple  or  com- 
plex problem,  depending  on  the  character  of  the  topography, 
and  more  especially  on  the  direction  of  the  streams  and  ridges 
as  compared  with  the  general  direction  of  the  proposed  road. 

3.  A  road   running  along  a  water-course  is  most  easily 
located.     In  this  case  the  choice  is  to  be  made  merely  between 
the  two  banks  of  the  stream,  or  between  keeping  one  bank 
continuously  and  making  occasional  crossings.      "When    the 
stream  is  small  it  will  usually  be  found  best  to  cross  it  at 
intervals,  the  advantage  of  direct  alignement  outweighing  the 
cost  of  bridging;  but  when  the  stream  is  of  considerable  size 
the  solution  of  the  problem  is  not  so  obvious,  requiring  patient 
comparison  of  results  in  the  two  cases  to  determine  whether  to 
cross  or  not,  while  in  the  case  of  the  larger  rivers  crossing 
may  be  out  of  the  question. 

When  there  is  a  choice  of  sides,  both  banks,  should  be 
traversed  by  the  engineer  on  reconnoissance,  and  while  exam- 
ining in  detail  the  one  side  he  should  take  a  general  and  com- 
prehensive view  of  the  other.  Only  thus  can  he  gain  a  complete 
knowledge  of  either  side.  The  points  to  be  considered  are  the 
relative  value  of  the  property  on  either  side,  the  number  and 


FIELD   ENGINEERING. 

size  of  tributary  streams,  and  probable  cost  of  crossing  them, 
the  cost  of  graduation  as  affected  by  the  amount  and  character 
of  the  material  to  be  removed,  and  the  liability  to  land  slides, 
the  amount  and  degree  of  curvature  required,  and  the  proba- 
ble revenues  which  the  road  can  command  If,  in  respect  to 
these  points,  one  bank  of  the  stream  gives  the  more  favorable 
result  all  the  "way,  the  question,  is  decided  at  once;  but  in 
case  the  greater  inducements  are  found  on  either  bank  alter- 
nately, as  usually  happens,  the  propriety  of  bridging  the 
stream,  with  the  costs  and  advantages,  must  be  considered  as 
an  additional  element  in  the  problem. 

4.  When  no  water-course  offers  along  which  the  road  may 
be  located,  the  difficulties  of  selecting  a  route  are  increased, 
and  these  usually  become  greatest  when  the  streams  are  found 
to  run  about  at  right  angles  to  the  direction  of  the  road.     Val- 
leys and  ridges  are  to  be  crossed  alternately,  involving  the 
necessity  of  ascending  and  descending  gradejs,  diverting  the 
road  from  a  straight  line,  and  increasing  the  distance  and  cur- 
vature.    The  engineer  must  now  seek  the  lowest  points  on  the 
ridges,  and  the  highest  banks  at  the  stream  crossings,  in  order 
to  reduce  as  much  as  possible  the  total  rise  and  fall,  but  these 
points  must  be  so  chosen  relatively  to  each  other  as  to  admit 
of  their  being  connected  by  a  grade  not  exceeding  the  maxi- 
mum which  may  be  allowable.      The  intervening  country 
between  summit  and  stream  must  usually  be  carefully  exam- 
ined, even  on  reconnoissance,  to  determine  where  the  assumed 
grade  will  find  sustaining  ground  at  a  reasonable  expense  for 
graduation  and  right  of  way. 

„  In  selecting  stream  crossings,  regard  should  be  had  not  only 
to  the  height  of  the  bank,  but  also  to  the  character  of  the  bot- 
tom, its  suitability  for  foundations,  and  its  liability  to  be 
washed  by  the  current.  The  direction  and  force  of  the  cur- 
rent should  be  observed,  and  its  behavior  during  freshets,  and 
the  extremes  of  high  and  low  wrater  ascertained,  if  possible. 
An  approximate  estimate  of  the  cost  of  bridging  may  be  made. 

5.  The  engineer  should  not  only  seek  the  best  ground  on  the 
route  first  assumed,  but  should  have  an  eye  to  all  other  possi- 
ble routes,  holding  them  in  consideration  pending  his  accu- 
mulation of  evidence,  and  being  ready,  finally,  to  adopt  that 
one  which  promises  the  greatest  ultimate  economy.    He  should 
be  able  to  read  the  face  of  the  country  like  a  map,  and  to 


RECCWNOISSANCE.  3 . 

carry  in  his  mind  a  continuous  idea  or  image  of  any  line  he  is  ex- 
amining, so  as  to  judge  with  tolerable  accuracy  of  the  influence 
any  one  portion  of  the  line  may  have  on  another  as  to  align- 
ment and  grade,  even  though  many  miles  apart.  In  the  success- 
ful prosecution  of  a  reconnoissance  he  must  depend  mainly  on 
his  own  natural  tact  and  a  judgment  matured  by  experience. 

6.  The  engineer  will  bring  to  his  aid  in  the  first  place  the 
most  reliable  maps,  and  those  drawn  on  the  largest  scale.  The 
sectional  maps  of  United  States  surveys  will  be  found  very 
useful  when  they  exist.  In  addition  to  these  it  is  often  desira- 
ble to  prepare  a  map  on  a  scale  of  one  or  two  inches  to  a  mile, 
on  which  will  be  drawn  the  principal  features  of  the  country 
to  be  traversed,  such  as  streams,  roads,  towns,  and  the  princi- 
pal ridges,  if  known,  but  leaving  the  further  details  to  be  filled 
in  by  the  engineer  as  he  progresses.  Such  a  map  furnishes  a  cor- 
rect scale  for  his  sketches,  and  saves  much  valuable  time,  as  he 
has  only  to  sketch  what  the  map  does  not  contain,  and  occa- 
sionally to  make  corrections  when  he  finds  the  map  to  be  in 
error.  He  also  notes  on  the  map  the  governing  points  of  the 
route,  such  as  the  best  crossings  of  streams,  ridges,  or  other 
roads,  and  any  point  where  the  line  will  evidently  be  com- 
pelled to  pq,ss  He  may  then  indicate  the  route  by  a  dotted 
line  on  the  map  drawn  through  the  governing  points.  Having 
traversed  the  route  in  one  direction  he  should  retrace  his  steps, 
verifying  or  correcting  his  observations,  and  making  such 
further  notes  as  seem  important.  When  in  a  densely  wooded 
country,  with  but  few  openings,  it  may  be  impossible  for  him 
to  get  a  commanding  view  from  any  point  that  will  afford  him 
the  necessary  information  as  to  the  general  topography.  He 
must  then  depend  largely  upon  instrumental  observations, 
taking  these  more  frequently,  and  noting  carefully  all  details 
likely  to  prove  useful  in  future  surveys. 

7.  The  instruments  required  on  an  extended  recon- 
noissance are  the  barometer  and  thermometer,  the  hand  or 
Locke  level,  a  pocket  or  prismatic  compass,  and  fl  telescope  or 
strong  field-glass.  To  these  may  be  added  a  telemeter  for 
measuring  distances  at  sight,  but  when  good  maps  are  to  be 
had  this  instrument  is  seldom  needed.  So  also  some  portable 
astronomical  instruments  are  necessary  in  a  new  country,  for 
determining  latitude  and  longitude,  but  would  only  be  a  use- 
less iucumbrance  in  a  settled  district. 


4  FIELD   ENGINEERING. 

8.  The  mercurial  barometer  has  generally  been  relied  upon 
for  the  determination  of  heights,  but  owing  to  its  inconvenient 
dimensions  and  the  danger  of  breaking,  it  is  now  discarded  by 
railroad  engineers   in  favor  of    the  more  portable   aneroid 
barometer,  except  in  the  case  of  trans-continental  surveys, 
and  when  astronomical  instruments  are  to  be  used  also. 

9.  The  best  aneroids  are  designed  to  be  self  compen- 
sating for  temperature,   so  that  with  a  constant  atmospheric 
pressure  the  reading  shall  be  the  same  at  all  temperatures  of  the 
instrument.    This,  however,  being  a  very  delicate  adjustment, 
is  not  always  successfully  made,  so  that  each  instrument  is  lia- 
ble to  have  a  small  error  due  to  temperature  peculiar  to  itself. 
This  error  will  be  found  rarely  to  exceed  one  hundredth  of  an 
inch,  plus  or  minus,  per  change  of  ten  degrees  Fah.,  and  is 
frequently  much  less  than  this.     Just  wliat  the  error  is  in  a 
particular  instrument  may  be  determined  by  careful  compari- 
son with  a  standard  mercurial  barometer  at  the  extremes  of 
temperature,  assuming  the  error  found  as  proportional  to  the 
diiference  of  temperature  for  all  intermediate  degrees  of  heat. 
The  error  having  been  determined  for  any  aneroid,  it  should 
be  applied,  with  its  proper  sign,  to  every  reading  to  obtain 
the  true  reading. 

The  sizes  generally  used  are  If  and  2|  inches  in  diameter, 
respectively,  and  experience  seems  to  prove  that  there  is  no 
advantage  in  using  larger  sizes,  but  rather  the  contrary. 

1C.  The  ordinary  barometric  formula3  and  tables  have  been 
prepared  with  reference  to  the  mercurial  barometer.  In  order 
that  they  may  apply  to  the  aneroid,  it  is.  necessary  that  the 
latter  should  be  adjusted  to  read  inches  of  mercury  identically 
with  the  mercurial  column  at  the  sea  level  at  a  temperature  of 
32°  Fah.  But  as  the  aneroid,  unlike  the  mercurial  column, 
requires  no  correction  for  latitude,  nor  for  the  variation  in  the 
force  of  gravity  due  to  elevation,  that  portion  of  the  formula 
which  provides  for  such  corrections,  as  well  as  that  which 
provides  for  a  correction  due  to  the  temperature  of  the 
instrument  itself,  may  be  omitted  when  using  an  aneroid. 
Thus  the  general  formula  is  very  much  simplified,  and  be- 
comes 

,  =  log  £  608818 '<    .<<  +  <'- 64° 


h'  \  900 


KECOKKOISSA^CE.  5 

in  which  7i,  and  h'  are  the  readings  of  the  aneroid  in  inches, 
and  tt  and  t'  the  readings  of  a  Fahrenheit  thermometer  at  the 
lower  and  upper  of  any  two  stations  respectively,  and  z  is  the 
difference  in  elevation  in  English  feet  of  those  stations. 

To  facilitate  the  calculation  of  heights  by  this  formula,  we 
may  write 

Log  |j  60384.3  =  [log  h,  -  log  h1]  60384.3 

and  since  only  the  difference  of  the  logs,  is  required,  this  will 
not  be  affected,  if  we  subtract  unity  from  each.  The  quan- 
tities in  Table  XV.  are  prepared,  therefore,  by  the  formula 

(log  7i  —  1)60384. 3 
for  every  ^ths  of  an  mcn  from  19  inches  to  31  inches. 

Table  XVI.  contains  values  of  -i^t  -  for  every  de- 

yUU 

gree  of  (t,  +  t')  from  20°  to  200°  Fah. 

11.  To  find  the  difference  in  elevation  of  any  two  stations  by 
the  tables : 

Take  the  difference  of  the  quantities  corresponding  to  Ti,  and 
7i'  in  Table  XV.  as  an  approximation,  and  for  a  correction 
multiply  this  difference  by  the  coefficient  corresponding  to 
(£,-{-  0,  in  Table  XVI.,  adding  or  subtracting  the  product 
according  to  the  sign  of  the  coefficient. 

Example. — 

Lower  Sta.  Upper  Sta. 

in.  in. 

Aneroid  h,  =  29.92  h'  =  23.57 

Thermometer  t,   =  77°.6  t  =  70°.4 

By  Table  XV.  for  29.92  we  have  28741 

for  23.57  22485 


Difference    6256 

By  Table  XVI.  for  77.6  +  70.4  =  148  we  have  -f  .0933 
Then  6256  X  .0933  =  583.6848 

and  6256  +     584  =  6840  ft.  =  z.—Ans. 

12.  Certain  precautions  are  to  be  observed  in  the  use  of  the 
aneroid.  When  the  index  has  been  adjusted  to  a  correct 
reading  by  means  of  the  screw  at  its  back,  it  should  not  be 
meddled  with  until  it  can  again  be  compared  with  a  standard 
mercurial  barometer,  and  even  then  some  engineers  prefer  to 
take  note  of  its  error,  if  any,  rather  than  disturb  the  aneroid. 


6  FIELD   ENGINEERING. 

Again,  since  the  principle  of  compensation  supposes  the 
aneroid  to  have  a  uniform  temperature  throughout  its  parts,  it 
must  be  guarded  against  sudden  changes,  as  otherwise  the 
metallic  case  will  be  considerably  heated  or  cooled  before  the 
change  can  affect  the  inner  chamber,  thus  inducing  very  erro- 
neous results.  The  aneroid,  therefore,  should  seldom  be  taken 
from  its  leather  case,  nor  exposed  to  any  radiant  heat  of  sun 
or  fire,  nor  worn  so  near  the  person  as  to  increase  its  tempera- 
ture above  that  of  the  surrounding  atmosphere.  If  removed 
to  an  atmosphere  of  decidedly  different  temperature,  time 
must  be  allowed  for  the  aneroid  to  be  thoroughly  permeated 
by  the  new  degree  of  heat.  The  aneroid  should  be  held  with 
the  face  horizontal  while  being  read ;  it  should  be  handled  care- 
fully, and  all  concussions  avoided,  and  it  should  be  compared 
with  a  standard  as  often  as  practicable  to  make  sure  that  it 
has  suffered  no  derangement.  Observing  these  precautions, 
and  having  a  really  good  aneroid,  the  engineer  should  obtain 
excellent  results  in  the  estimation  of  heights.  It  has  been 
found  that  the  slight  error  in  compensation,  previously  alluded 
to,  is  subject  to  a  change  during  the  first  year  or  two  after  the 
instrument  is  made,  but  subsequently  it  becomes  quite  per- 
manent. 

13.  For  the  purpose  of  obtaining  approximate  elevations  by 
a  simple  inspection  of  the  dial,  the  modern  aneroid  is  provided 
with  a   secondary  scale  reading  hundreds  of  feet,  which  is 
placed  outside  the  scale  of  inches.     It  is  divided  according  to 
the  following  formula  prepared  by  Prof.  Airy : 

s  =  65033^(1  +  ^--) 

in  which  it  is  evident  that  no  correction  for  temperature  is 
required  when  the  average  temperature  of  the  two  stations  is 
50°.  When  the  two  scales  are  engraved  on  the  same  plate  the 
zero  of  the  scale  of  feet  is  coincident  with  31  on  the  scale  of 
inches;  but  in  some  aneroids  the  scales  are  on  two  concentric 
plates,  so  that  the  zero  of  one  may  be  made  to  coincide  with 
any  division  of  the  other,  which  is  in  some  respects  an  advan- 
tage. 

14.  The  theory  of  the  barometer,  as  expressed  in  the  above 
formula,  assumes  the  atmosphere  to  be  at  rest,  and  its  pres- 
ure  affected  only  by  temperature,  whereas,  in  fact,  the  pres- 


RECOK^OISSAKCE.  7 

sure  at  any  point  is  liable  to  sudden  changes  due  to  variations 
in  the  force  of  the  wind,  the  amount  of  humidity,  etc.  The 
best  way  to  eliminate  errors  due  to  these  causes  is  to  take  read- 
ings simultaneously  at  the  points  the  elevations  of  which  are 
to  be  compared.  For  this  purpose  an  assistant  should  be 
stationed  at  some  point  of  known  elevation  contiguous  to  the 
route  to  be  surveyed,  and  provided  with  an  aneroid  similar  to 
that  carried  by  the  engineer.  The  aneroids,  time-pieces,  and 
thermometers  having  been  compared  at  this  point,  the  assist- 
ant should  record  the  readings  every  ten  minutes,  with  the 
time,  temperature,  and  state  of  the  weather.  The  engineer 
will  thus  have  a  standard  with  which  to  compare  his  own 
observations.  If  the  survey  is  so  extended  that  the  same  con- 
ditions of  atmosphere  are  not  likely  to  be  experienced  by  the. 
two  observers,  the  assistant  should  be  instructed  to  move  for- 
ward to  a  new  station  at  a  designated  time;  or  two  assistants 
may  be  employed,  one  at  each  of  two  stations  between  which 
the  engineer  intends  to  make  a  reconnoissance.  Even  with 
these  precautions  no  attempt  should  be  made  to  obtain  the  ele- 
vation of  important  points  during,  or  just  before,  or  after  a 
storm  of  wind  or  rain. 

15.  When  but  one  aneroid  is  used  the  observations  at  the 
several  stations  should  be  taken  as  nearly  together  as  possible 
in  point  of  time,  and  then  repeated  in  inverse  order,  taking 
the  mean  of  the  observations  at  each  station,  and  repeating  the 
whole  operation  if  necessary.  Only  approximate  results  can 
be  hoped  for,  however,  with  a  single  instrument,  unless  the 
atmospheric  conditions  are  very  favorable. 

10.  The  Locke  Level  is  an  instrument  in  which  the 
bubble  and  the  observed  object  may  be  seen  at  the  same  instant, 
enabling  the  operator  to  keep  the  instrument  horizontal,  while 
holding  it  in  the  hand,  like  an  ordinary  spy-glass.  While 
very  portable,  it  enables  the  observer  to  define  rapidly  all  visi- 
ble points  of  the  same  elevation  as  his  own,  and  to  estimate 
from  these  the  relative  heights  of  other  points.  It  may  be 
made  useful  in  a  variety  of  ways  which  easily  suggest  them- 
selves to  the  engineer  in  cases  where  no  great  precision  is 
required,  and  where  a  more  elaborate  level  is  not  at  hand. 

17.  The  Prismatic  Compass  is  a  portable  instrument 
with  folding  sights,  in  using  which  the  bearing  to  an  object 
may  be  read  at  the  same  instant  that  the  object  is  observed. 


8  FIELD   ENGINEERING.       ^ 

The  bearings  are  read  upon  a  floating  card,  graduated  and 
numbered  from  zero  to  360°,  so  that  no  error  can  be  made  in 
substituting  one  quadrant  for  anotlier.  The  instrument  may  be 
held  freely  in  the  hand  during  an  observation,  though  better 
results  are  obtained  by  giving  it  a  firm  rest. 


CHAPTER  II. 
PRELIMINARY  SURVEY. 

18.  A  preliminary  survey  consists  in  an  instrumental  exam- 
ination of   the  country  along  the  proposed   route,   for  the 
purpose  of  obtaining   such  details  of   distances,   elevations, 
topography,  etc. ,  as  may  be  necessary  to  prepare  a  map  and 
profile  of  the  route,  make  an  approximate  estimate  of  the  cost 
of  constructing  the  road,  and  furnish  the  data  from  which  to 
definitely  locate  the  line  should  the  route  be  adopted.     The 
survey  is  more  or  less  elaborate,  according  to  circumstances. 
In  case  the  country  is  new,  or  the  reconnoissance  has  been 
incomplete,  or  if  several  routes  seem  to  offer  almost  equal 
inducements,  the  survey  wrill  partake  somewhat  of  the  nature 
of  a  reconnoissance,  and  will  be  made  more  hastily  than  if  but 
one  route  is  to  be  examined,  and  that,  perhaps,  presenting 
serious  engineering  difficulties.     The  survey  is  made  as  expe- 
ditiously  as  possible,  consistent  with  general  accuracy,  but 
should  not  usually  be  delayed  for  the  sake  of  precision  in 
matters  of  minor  detail. 

19.  For  preliminary  survey  the  Corps  of  engineers  is 
organized  as  follows: 

A  chief  engineer,  an  assistant  engineer,  two  chainmen,  one 
or  two  axemen,  a  stakeman,  and  a  topographer,  these  forming 
the  compass  (or  transit)  party,  to  which  a  flagman  is  some- 
times added;  a  leveller  and  one  or  two  rodmen,  forming  the 
level  party;  and  to  these  is  sometimes  added  a  cross  level  party 
of  two  or  three  assistant  rodmen. 

20.  The  chief  engineer  takes  command  of  the  corps, 
and  directs  the  survey.     He  ascertains  or  estimates  the  value 
of  the  lands  passed  over,  the  owners'  names,  and  the  boundary 
lines  crossed  by  the  line  of  survey.     He  examines  all  streams, 


PRELIMINARY    SURVEY.  9 

and  estimates  the  size  and  character  of  the  culverts  and 
bridges  which  they  will  require;  he  notices  existing  bridges, 
and  inquires  concerning  their  liability  to  be  carried  away  by 
freshet;  he  selects  suitable  sites  for  bridges,  examines  the 
character  of  the  foundations,  the  direction  of  the  current  rela- 
tively to  that  of  the  line,  and  considers  any  probable  change 
in  the  direction  of  the  current  during  freshets;  he  inspects  the 
various  soils,  rocks,  and  kinds  of  timber  as  they  are  met  with, 
and  takes  full  notes  of  all  these  and  kindred  items  in  his  field 
book.  He  not  unfrequently  assumes  in  addition  the  duties  of 
topographer.  He  should  run  his  line  as  nearly  as  may  be  over 
the  ground  likely  to  be  chosen  for  location,  so  that  the  infor- 
mation obtained  may  be  pertinent,  and  so  that  the  length  of 
the  line,  the  shape  of  the  profile,  and  the  estimate  based  on 
the  survey  may  approximate  to  those  of  the  proposed  location. 
To  this  end  he  has  due  regard  to  the  levels  taken,  and  when 
they  show  that  the  line  as  run  fails  to  be  consistent  with 
allowable  grades,  he  either  orders  the  corps  back  to»  some 
proper  point  to  begin  a  new  line,  or  makes  an  offset  at  once 
to  a  better  position,  or  continues  the  same  line  with  some 
deflection,  simply  noting  the  position  and  probable  elevation 
of  better  ground,  as  in  his  judgment  he  thinks  "best.  He 
should  at  all  times  maintain  a  friendly  attitude  toward  pro- 
prietors, and  by  his  polite  bearing  endeavor  to  secure  their 
cordial  support  of  the  new  enterprise.  If  he  is  tolerably  cer- 
tain that  the  location  will  follow  nearly  the  line  of  the  prelim- 
inary survey,  he  should  have  with  him  some  blank  deeds  of 
right  of  way,  and  let  these  be  signed  by  land-owners  while 
they  are  favorably  disposed.  When  this  cannot  be  done,  a 
blank  form  of  agreement  to  allow  the  surveys  and  construc- 
tion of  the  road  to  proceed  until  such  time  as  the  terms  of 
right  of  way  may  be  agreed  upon  may  be  made  very  useful. 
The  chief  also  selects  quarters  for  his  men,  and  in  case  of 
camping  out  he  directs  the  movements  of  the  camp  equipage. 
21.  The  assistant  engineer  takes  the  bearings  of  the 
courses  run,  and  makes  a  minute  of  them,  with  their  lengths,  or 
the  numbers  of  the  stations  where  they  terminate.  He  sees  that 
the  axemen  keep  in  line  while  clearing,  and  the  chainmen 
while  measuring;  he  takes  the  bearings  of  the  principal  roads 
and  streams,  and  of  property  lines  when  met  with.  In  an 
open  country  he  may  save  time  by  selecting  some  prominent 


10  FIELD 

distant  object  toward  which  the  chainmen  measure  without  his 
assistance,  while  he  goes  forward  and  prepares  to  take  the 
bearing  of  the  course  beyond.  In  traversing  a  forest  with  not 
too  dense  undergrowth,  when  the  line  is  being  run  to  suit  the 
ground  according  to  a  given  grade,  it  i^  a  good  plan  for  the 
assistant  to  go  ahead  of  the  chainmen  as  far  as  he  can  be  seen, 
select  his  ground,  take  his  bearing  by  backsight  on  the  last 
station,  and  then  have  the  chainmen  measure  toward  him.  In 
this  case  both  he  and  the  head  chainman  should  be  provided 
with  a  good  sized  red  and  white  flag,  mounted  on  a  straight 
pole,  to  be  waved  at  first  to  call  attention,  and  afterward  held 
vertically  for  alignement.  Otherwise  a  flagman  must  be  added 
to  the  party,  who  will  select  the  ground  ahead,  under  the  in- 
structions of  the  chief,  and  toward  whom  the  survey  will  pro- 
ceed in  the  usual  manner. 

22.  The  head  chainmaii  drags  the  chain,  and  carries  a 
flag  which  is  put  into  line  at  the  end  of  each  chain  length  by  the 
assistant  engineer  or  the  rear  chainman.     It  is  his  duty  to 
know  that  his  flag  is  in  line  and  that  his  chain  is  straight  and 
horizontal  before  making  any  measurement,  and  to  show  the 
stakeman  where  each  stake  is  to  be  driven.    A  stake  is  usually 
driven  at  the  end  of  each  measured  chain  length,  called  a 
station,  though  in  an  open  and  level  country  the  stakes  at  the 
odd  stations  may  be  omitted,  in  which  case  marking  pins  are 
used  to  indicate  the  odd  stations  temporarily.     In  case  there 
is  much  clearing  to  be  done  the  head  chainman  plants  his  flag 
in  line,  and  ranging  past  it,  indicates  to  the  axemen  what  is 
to  be  cut,  going  a  little  in  advance  through  the  bushes  so  that 
they  may  work  toward  him.     The  head  chainman  should  be  a 
quick,  active  and  strong  man,  with  a  good  eye  and  a  taste  for 
his  work,  as  very  much  of  the  real  progress  of  the  survey 
depends  upon  him. 

23.  The  rear  chainman  holds  his  end  of  the  chain  firm- 
ly at  the  last  stake  or  pin  by  his  own  strength,  not  by  means  of 
the  stake.    He  keeps  the  tally  by  the  pins  when  they  are  used, 
and  watches  the  numbers  on  the  stakes  to  see  that  they  are  cor- 
rect.    The  end  of  a  course  should  always  be  chosen  at  the  end 
of  a  chain,  if  possible,  and  if  not,  then  at  a  brass  tag  indicating 
tens  of  feet,  as  thus  the  labor  of  plotting  the  map  will  be  much 
lessened.     The  numbering  of  stations  is  not  recommenced 
with  each  new  course,  but  is  continued  from  the  beginning  to 


PRELIMINARY    SURVEY.  11 

the  end  of  the  survey,  through  all  its  courses,  and  if  one 
course  ends  with  a  portion  of  a  chain  the  next  course  begins 
with  the  remainder  of  it.  It  is  the  rear  chainman's  duty  to 
attend  to  this,  holding  the  proper  link  at  the  compass  station. 
Any  fraction  of  a  chain  measured  on  the  line  is  called  a  plus, 
and  is  counted  in  feet  from  the  previous  station.  The  length 
of  an  offset  in  the  line  is  never  included  in  the  length  of  the 
line,  but  if  the  line  should  change  its  course  by  a  right  angle, 
or  more,  or  less,  the  numbering  would  go  on  as  usual. 

24.  The  axemen  should  be  accustomed  to  chopping  and 
clearing,  and  are,  therefore,  to  be  selected  in  the  country  rather 
than  the  city.     They  will  cut  out  so  much  of  the  underbrush 
and  overhanging  branches  as  may  interfere  with  the  sight  of 
the  assistant  or  leveller;  but  care  must  be  taken  not  to  cut 
unnecessarily  wide,  and  no  tree  of  considerable  size  should  be 
felled,  except  in  rare  instances.     When  running  by  compass,  if 
the  assistant  goes  ahead  of  the  chain,  he  can  always  select  a 
position  so  that  no  large  tree  will  interfere ;  or,  if  the  line  must 
be  produced  and  strikes  a  tree,  the  compass  may  be  brought  up 
and  set  close  to  the  tree  on  the  forward  side  as  nearly  in  line  as 
can  b3  estimated,  the  slight  error  in  offset  being  neglected, 
since  the  lino  will  1)3  produc3d  parallel  to  itself  by  the  needle. 

25.  The  stakemau  prepares  and  marks  the  stakes,  and 
drives  them  at  the  p.pints  indicated  by  the  head  chainman. 
When  no  clearing  is  needed,  the  axemen  keep  him  supplied 
with  stakes,  as  the  rapid  progress  of  the  chain  will  only  give 
him  time  to  drive  them.   The  stakes  should  be  two  feet  long  and 
pointed  evenly  so  as  to  drive  straight,  and  are  blazed  or  faced 
on  two  opposite  sides,  one  of  which  is  marked  in  red  chalk 
with  the  number  of  the  station.     The  stake  must  be  driven 
vertically,  and  with  the  marked  face  to  the  rear,  so  that  it  may 
be  read  by  the  <*odman  as  he  follows  the  line. 

26.  The  topographer  makes  accurate  sketches  of  all 
features  of  the  country  immediately  on  the  line,  and  extends 
the  sketches  as  far  each  side  of  the  line  as  he  can,  in  a  book 
prepared  for  the  purpose.     He  must  never  sketch  in  advance 
of  the  chain,  nor  in  advance  of  his  own  position.     His  work 
should  be  done  to  scale  as  nearly  as  possible,  using  the  same 
scale  for  distances  on  the  line  and  at  right  angles  to  it.     The 
scale  adopted  should  never  be  less  than  that  of  the  map  to  be 
made  from  the  sketches.     The  ruled  lines  of  a  field  book  are 


12  FIELD   ENGINEERING. 

usually  one  quarter  of  an  inch  apart,  so  that  a  scale  of  one 
line  to  a  station  equals  about  four  hundred  feet  to  an  inch. 
This  is  the  smallest  scale  ever  used.  The  scale  of  two  lines  to 
a  station  is  most  convenient  for  general  use.  Four  lines  to  a 
station  are  needed  in  special  cases  to  show  details,  as  in  pass- 
ing through*  villages.  The  scale  may  be  changed  from  time  to 
time  as  found  necessary,  but  no  two  scales  should  ever  be  used 
on  the  same  page.  The  numbering  of  the  stations  up  the  page 
indicates  the  scale  of  the  sketch. 

27.  When  the  contours  of  the  surface  are  required,  the 
topographer  may  join  the  level  party  in  order  that  his  esti- 
mates of  heights  and  slopes  may  be  corrected  by  the  instru- 
ment. He  should  never  draw  a  mass  of  contours  indiscrimi- 
nately, but  should  sketch  them  as  they  exist  at  a  uniform  ver- 
tical interval.  This  interval  may  be  assumed  at  five  feet  in 
a  gently  rolling  country,  and  at  twenty  feet  in  a  mountainous 
one,  but  an  interval  of  ten  feet  will  be  found  most  convenient 
generally.  If  the  topographer  accompanies  the  level  he  can 
assume  the  contours  at  the  even  tens  of  feet  in  elevation,  and 
mark  them  so,  noting  where  a  contour  crosses  the  surveyed 
line,  and  sketching  its  direction  and  shape  both  ways  from 
that  point.  He  will  estimate  the  rate  of  slope  of  the  ground 
at  right  angles  to  the  line  as  so  many  feet  per  hundred,  and 
record  it  from  time  to  time,  noting  ascent  from  the  line  on 
either  side  by  "A,"  and  descent  by  "  D."  If  the  slope  changes 
within  the  limit  of  the  page,  the  line  of  change  may  be 
sketched  and  the  next  slope  recorded.  When  little  banks  or 
terraces  occur,  or  bluffs  and  rocks,  which  cannot  be  suf- 
ficiently indicated  by  contours,  they  should  be  shown  by 
hatchings,  and  the  height  noted.  Special  care  should  be 
taken  to  sketch  roads  and  houses  in  their  correct  positions 
and  dimensions,  the  latter  to  bo  either  measured,  paced  or 
estimated.  The  dimensions  should  also  be  recorded  in  num- 
bers. The  outline  of  forests  may  be  shown  by  a  scalloped 
line,  and  the  kind  of  timber,  and  whether  dense  or  scattered, 
written  within  the  inclosed  space.  Correct  outlines  are  essen- 
tial, but  no  time  should  be  given  to  shading  up  a  sketch  with 
conventional  signs.  A  single  sign,  or  the  name  of  the  thing 
intended,  is  all  sufficient.  Land-owners'  and  residents'  names 
should  be  recorded  whenever  they  can  be  obtained,  as  well  as 
the  names  of  roads,  streams  and  public  buildings. 


PRELIMINARY    SURVEY.  13 

28.  The  leveller  takes  charge  of  the  level  party  and 
keeps  the  notes  of  his  work.     He  reads  the  rod  on  benches  and 
at  turning  points  to  hundredths  of  a  foot  and  to  tenths  at  other 
points.     He  should  direct  a  bench  to  be  made  at  least  once 
every  half  mile,  and  in  a  very  rough  country  every  quarter  of 
a  mile.     The  benches  need  not  be  far  from  the  line,  and,  if 
well  chosen,  may  be  used  as  turning  points,  thus  saving  time. 
The  elevation   of  turning  points  must  be   computed   when 
taken,   so  that  the  elevation  of  any  one   of  them  may  bo 
instantly  given  when  called  for,  and  the  other  elevations  will 
be  filled  in  as  far  as  may  be  without  delaying  the  survey.     As 
the  levels  are  usually  the  most  essential  part  of  the  survey, 
much  care  should  be  taken  to  have  the  instrument  well  ad- 
justed and  truly  level,  and  the  rod  held  vertically  and  correctly 
read  on  turning  points,  but  the  intermediate  work  should  not 
be  so  done  as  to  delay  the  party  unnecessarily.     The  leveller 
should  use  every  endeavor  to  follow  closely  after  the  survey- 
ing party,  so  that  the  chief  and  topographer  may  have  the 
advantage  of  his  notes. 

29.  The  rodman's  first  duty  is  to  hold  the  rod  vertically, 
and  he  must  learn  to  do  this  in  calm  or  windy  weather,  in  level 
field  or  on  side  hill.     He  may  carry  a  small  disk-level,  which 
applied  to^he  edge  of  the  rod  will  show  when  it  is  vertical. 
'The  turning  points  are  to  be  selected  for  firmness  and  definite- 
ness,  and  so  that  they  will  afford  a  clear  view  from  beyond 
for  a  backsight.     The  rod  is  held  for  a  reading  on  the  ground 
at  every  stake,  the  number  of  which  is  called  out  to  the  level- 
ler as  soon  as  the  rodman  arrives  at  it ;  the  rod  is  also  to  be 
held  at  every  prominent  change  of  slope  on  the  line,  as  the 
crest  and  foot  of  every  bank,  the  rodman  calling  out  its  dis- 
tance from  the  last  stake  as  plus  so  many  feet,  but  all  gentle 
undulations  and  minor  irregularities  are  to  be  neglected.    The 
rod  will  always  be  read  at  the  surface  of  a  stream  or  pond, 
and  also  at  its  deepest  part  on  the  line,  when  possible;  other- 
wise the  depth  of  the  water  may  be  found  by  sounding,  and 
so  recorded.     Should  the  line  run  along  a  stream  the  surface 
will  be  taken  occasionally,  opposite   certain  stations,  and  in 
case  of  a  canal,  the  elevation  of  surface  above  and  below  each 
lock  must  be   noted.     The  rodman  makes  inquiry  for  high- 
water  marks  or  seeks  traces  of  them  himself  in  an  uninhabited 
district,  and  holds  the  rod  upon  them  that  their  elevation  may 


14  FIELD   ENGINEERING. 

be  determined.  The  rodman  carries  a  small  axe  or  hatchet 
with  which  to  make  benches  and  to  trim  out  any  stray 
branch  that  may  intercept  the  leveller's  view. 

30.  The  assistant  rodmen  take  the  slope  and  elevations 
of  the  ground  at  right  angles  to  the  line,  using  vertical  and  hori- 
zontal rods  and  a  pocket  level,  or  a  tape  line  and  clinometer. 
The  cross  levels  are  not  taken  throughout  the  whole  survey,  if 
at  all,  but  only  where  the  roughness  of  the  country  seems  to 
demand  it.     They  may  be  extended  to  any  distance  from  the 
centre  line  required  by  the  chief — not  less,  however,  than  fifty 
feet  as  a  rule.     They  may  be  taken  at  the  stations  only,  or 
oftener,  if  necessary,  depending  upon  the  roughness  of  the 
surface,  the  object  being  to  define  accurately  the  contours, 
and  so  the  shape  of  the  ground.     The  assistant  rodmen  will 
also  take  soundings  when  they  are  needed,  either  on  the  line 
or  at  right  angles  to  it. 

31.  In  defining  the  duties  of  the  members  of  the  corps,  the 
instruments  used  have  been  incidentally  noticed. 

32.  The  compass  is  preferable  to  the  transit  on  prelimi- 
nary surveys,  because  it  can  be  operated  more  rapidly,  is  lighter, 
and  usually  has  a  better  needle.     It  may  have  either  plain 
sights  or  telescope,  and  be  mounted  on  tripod  or  Jacob  staff. 
The  simpler  forms  are  preferred  for  forest  work.     Not  unfre- 
quently  the  engineer's  transit  is  employed,  but  usin°fthe  needle. 
A  preliminary  line  should  not  be  run  by  backsights  and  deflec- 
tions, unless  local  attraction  is  found  to  exist  to  such  an  extent 
as  to  destroy  confidence  in  the  needle;  or,  in  special  cases, 
where  the  natural  obstacles  to  a  survey  are  very  great.     In  the 
latter  case  the  survey  partakes  of  the  nature  of  a  location,  and 
should  be  conducted  with  similar  care  and  fidelity. 

33.  The   chain   is  100  feet  long,  and  composed  of  100 
links.    It  should  be  of  steel  for  lightness,  durability,  and  greater 
accuracy.      Those  having  rings  of   hard  steel,  unbrazed,  are 
least  apt  to  wear.     Five  marking  pins  are  needed,  each  having 
a  piece  of  red  flannel  attached,  for  temporal y  stations,  or  for 
keeping  points  temporarily  while  measuring  by  parts  of  a 
chain  up  or  down  a  slope.     A.  pointed  plumb  bob,  with  sev- 
eral yards  of  small  cord  wound  on  a  carpenter's  spool,  is  use- 
ful in  chaining  over  steep  declivities  or  bluffs. 

34.  The  axes  should  be  of  best  quality,  with  hand-made 
handles,  and  not  too  heavy.     The  axe  of  the  stakeman  should 


PRELIMINARY    SURVEY.  15 

have  a  fine  edge  for  dressing  and  a  broad  head  for  driving 
the  stakes.  When  the  stakes  are  not  required  to  be  over  two 
feet  long,  a  stout  basket,  having  a  square,  flat  bottom,  26x14 
inches,  should  be  furnished  the  stakeman.  He  will  then  pre- 
pare a  basketful  of  stakes,  ready  marked,  and  place  them  in 
the  basket  regularly,  in  the  reverse  order  of  their  numbers,  so 
that  they  will  couie  to  hand  as  wanted.  A  small  hand-saw 
no  larger  than  the  basket,  with  rather  coarse  teeth,  wide  set, 
will  be  found  extremely  useful  in  cutting  stakes  with  square 
heads  and  of  uniform  length,  and  much  more  rapidly  than  can 
be  done  with  an  axe.  When  not  in  use,  it  is  to  be  strapped  to 
the  inside  of  the  basket,  to  prevent  its  being  lost  by  the  way. 
When  the  basket  is  about  empty,  the  stakeman,  with  the 
assistance  of  the  axemen,  can  soon  replenish  it,  and  the  stakes 
being  all  numbered  at  once,  there  is  less  danger  of  a  mistake 
being  made  in  the  tally  than  when  they  are  marked  only  as 
wanted. 

35.  The  level  should  be  the  regular  engineer's  level,  the 
same  as  used  on  location. 

36.  The  rod  should  be  self -reading,  i.e.,  to  be  read  by  the 
leveller,  as  too  much  time  would  be  consumed  in  the  constant 
adjustment  of  a  target  by  the  rodman.    It  should  be  as  long  as 
can  be  conveniently  handled  in  order  to  reduce  the  number  of 
turning  points  on  hill  sides.     A  very  convenient  rod  may  be 
made  of  thoroughly  seasoned  clear  white  pine,  sixteen  feet 
long  and  two  inches  wide,  with  a  thickness  of  one  inch  at  the 
bottom,  increasing  to  one  and  a  quarter  inches  at  six  feet  from 
the  bottom,  and  then  gradually  diminishing  to  three  eighths  of 
an  inch  at  the  top.     The  rod  is  shod  with  a  stout  strap  of  steel, 
extending  five  inches  up  the  edges,  and  secured  by  screws. 
The  top  is  protected  for  a  few  inches  by  a  plate  of  sheet  brass 
on  the  back.     The  face  of  the  rod  is  a  plain  surface  through- 
out, and  is  graduated  from  the  lower  edge  of  the  steel  shoe  as 
zero.     The  divisions  are  fine  cuts  made  with  the  point  of  a 
knife.     At  the  foot  and  half-foot  points  the  cuts  extend  across 
the  face.     For  the  tenths  and  half  tenths  they  extend  three 
quarters  of  an  inch  from  the  right  hand  edge,  terminating  in  a 
line  scribed  parallel  to  the  edge  of  the  rod,  thus  forming  rec- 
tangular blocks  half  a  tenth  wide,  every  other  one  of  which  is 
painted  black,  the  body  of  the  rod  being  white.     The  feet  are 
indicated  by  numerals  painted  red  on  the  blank  part  of  the 


1G  FIELD 


face,  each  figure  standing  exactly  on  its  foot  mark,  and  being 
exactly  one  tenth  high.  For  the  figure  5  the  Roman  V.  is  sub- 
stituted and  for  9  the  Roman  IX.,  so  that  in  case  a  dumpy 
level  is  used  the  5  may  not  be  mistaken  for  a  3,  nor  the  9  for  a 
6.  At  the  half-foot  points  a  red  diamond  is  painted,  so  that 
the  graduated  line  bisects  it.  No  other  figures  nor  gradua- 
tions are  required.  With  this  rod  the  leveller  can  read  quite 
accurately  to  hundred  ths  of  a  foot,  and  after  some  practice 
can  estimate  the  half  hundredths. 

37.  The  horizontal  rod  for  cross-levels  maybe  made 
of  white  pine,  ten  feet  long  and  one  inch  thick  by  three  wide, 
tipped  with  brass,  painted  white,  and  graduated  to  feet  and 
tenths.    It  must  be  a  straight  edge,  and  is  levelled  by  a  pocket 
level  placed  upon  it  when  needed,  or  by  a  small  level  embedded 
permanently  in  one  edge.     The  vertical  rod  to  be  used  with  it 
is  made  of  pine  eight  feet  long  and  one  and  a  quarter  inches 
square,  and  graduated  to  feet,  tenths,  and  half  tenths.     All 
rods  when  not  in  use  should  be  laid  on  a  flat  surface  to  pre- 
vent their  being  sprung.     Leaning  them  in  a  corner  soon  ruins 
them  for  use. 

38.  The  clinometer  is  any  small  instrument  which  will 
measure  the  slope  angle  of  the  surface.     The  angle  is  always 
estimated  from  the  horizon,  a  vertical  being  90°.    The  rise  per 
100  feet  is  found  by  multiplying  the  nat.  tangent  of  the  slope 
angle  by  100.     It  may  often  be  found  more  easily  by  the 
leveller  reading  the  rod  at  a  station  and  then  100  feet  left  or 
right  of  the  line.     If  surface  measures  are  taken  in  connec- 
tion with  a  slope  angle  they  are  reduced  to  horizontal  meas- 
ures by  multiplying  them  by  the  cosine  of  the  slope  angle. 

39.  The  plane-table  is  rarely  if  ever  used  on  prelimi- 
nary surveys  in  the  United  States.     Occasional  bearings  taken 
to  prominent  objects  by  the  assistant  engineer,  or  the  use  of  a 
prismatic  compass  by  the  topographer  in  connection  with  his 
sketches,  is  found  to  answer  every  purpose. 

40.  In  case  a  survey  is  to  be  made  with  a  tran- 
sit, it  is  necessary  to  add  a  back  flagman  to  the  party,  who  will 
hold  his  flng  or  rod  on  the  point  last  occupied  by  the  transit,  so 
that  the  assistant  may  take  a  backsight  upon  it.     The  direction 
of  a  new  course  in  each  case  is  determined  by  the  deflection 
angle  to  the  right  or  left  of  the  preceding  course  produced. 
The  bearing  of  one  long  course  near  the  beginning  of  the  sur- 


PRELIMINARY    SURVEY.  17 

rey  having  been  carefully  ascertained,  the  bearing  of  each  suc- 
ceeding course  is  calculated  from  the  deflections,  and  entered 
in  a  column  of  the  field  book  headed  Calculated  Bearings,  from 
which  the  line  is  afterwards  plotted.  The  magnetic  bearing 
of  each  course  should  also  be  taken  from  the  needle,  and  re- 
corded as  such,  but  is  used  only  as  a  check  on  the  transit 
work.  The  deflections  should  be  made  in  degrees,  halves,  or 
quarters,  if  possible,  to  facilitate  the  calculation  of  bearings, 
and  to  admit  of  using  a  traverse  table. 

41.  The  attached  level  and  vertical  arc  of  the  transit  are 
useful  in  determining  approximately  the  grade  of  the  line  run 
in  advance  of  the  level  party,  or  in  seeking  for  one  assumed 
grade  to  which  it  is  desired  that  the  line  shall  conform.     For 
this  purpose  it  is  only  necessary  to  set  the  vertical  arc  to  the 
angle  corresponding  to  the  grade  as  given  in  Table  XIV.,  and 
let  the  head  chainman  move  about  until  a  point  on  his  rod  at 
the  same  height  from  the  ground  as  the  telescope  is  covered 
by  the  horizontal  cross-hair. 

42.  The  point  on  the  ground  where  a  transit  is  set  up  is 
marked  by  a  good-sized  plug,  flat  headed,  and  driven  down 
flush  with  the  ground,  with  a  tack  set  in  the  head  to  show  the 
exact  point  or  centre.    This  is  called  a  transit  point.     When 
a  transit  point  occurs  at  a  regula?  station,  the  stake  bearing 
the  number  of  that  station  is  set  three  feet  to  the  left  cf  the 
line  opposite  the  plug  and  facing  it.     When  a  transit  point 
occurs  between  stations  the  stake  is  driven  three  feet  to  the 
left  of  it,  marked  with  the  number  of  the  preceding  station 
-j-  the  distance  from  that  station  in  feet. 

43.  As  a  transit  is  capable  of  giving  a  line  with  great  pre- 
cision, it  is  important  that  the  flags  used  in  connection  with 
it  should  be  equally  precise  in  giving"points.    An  excellent  flag 
for  this  purpose  is  made  of  well-seasoned  clear  white  pine  ten 
feet  long,  two  and  a  half  inches  wide,  and  one  inch  thick.    It  is 
tapered  for  the  last  four  inches  to  an  ^ge  atone  end,  the  edge 
being  formed  at  the  middle  of  the  width.     The  tapered  end  is 
shod  with  a  band  of  steel  covering  the  edge  of  the  rod,  and 
secured  by  screws,  and  the  steel  is  brought  to  a  sharp  edge  at 
the  point  of  the  rod.     The  rod  is  then  painted  white  and 
tipped  with  brass  at  the  square  or  upper  end.     A  centre  line 
on  the  face  is  then  struck  from  the  point  of  the  steel  to  the 


18  FIELD   ENGIKEEKING. 

middle  of  the  brass  tip  by  means  of  a  piece  of  sewing  silk, 
and  a  fine  cut  made  with  a  knife  and  steel  straight  edge. 
The  centre  line  must  not  be  scribed  parallel  to  one  edge  of  the 
rod,  as  this  is  rarely  ever  straight.  The  face  of  the  rod  is 
then  divided  into  one-foot  spaces,  measured  from  the  head  of 
the  rod,  and  these  are  painted  red  on  either  side  of  the  centre 
line  in  alternate  blocks.  On  the  back  of  the  rod  at  three  and 
a  half  feet  from  the  point  is  placed  a  small  ground-glass 
bubble-tube,  mounted  very  simply,  and  attached  to  the  rod  by 
a  brass  plate  and  screws,  and  guarded  by  two  blocks  of  wood 
for  protection.  The  centre  line  of  the  rod  is  made  vertical  by 
a  plumb-line  while  the  level  tube  is  being  attached,  which  ever 
after  secures  a  vertical  rod.  If  only  two  feet  of  this  rod  can 
be  seen  over  any  obstruction,  a  point  can  be  set  with  great 
precision,  provided  the  level  tube  is  in  adjustment.  This  flag 
can  also  be  used  as  a  plumb  in  chaining  wTith  much  more 
satisfaction  than  a  cord  and  weight,  especially  in  windy 
weather. 

44.  A  transit  survey  usually  requires  more  clearing  than 
one  made  by  compass.    When  a  given  course  is  to  be  produced 
in  a  forest,  some  large  trees  will  inevitably  be  encountered,  but 
the  labor  and  delay  of  felling  them  may  be  avoided  by  the 
use  of  auxiliary  lines.     These  may  be  classified  as  running 
parallel  to  the  main  line,  at  a  small  angle  with  it,  or  at  a  large 
angle  with  it. 

45.  The  parallel  line  is  established  by  means  of  two 
short  perpendicular  offsets  measured  with  care  before  reach- 
ing the  obstacle,  and  the  main  line  is  established  beyond  the 
obstacle  by  means  of  two  more  equal  offsets.     But  since  short 
back-sights  are  to  be  avoided,  these  offsets  should  be  at  least 
100  feet  apart,  so  that  it  may  be  difficult  to  find  a  parallel  line 
of  sufficient  length  which  does  not  strike  some  other  obstacle, 
or  at  Jleast  require  considerable  extra  clearing. 

46.  The  auxiliary  lines  making-  a  small  angle 
with  the  main  line   are   more  convenient,  not  only  on   this 
account,  but  because  they  require  a  less  number  of  transit 
points.   By  them  an  isosceles  triangle  is  formed  on  the  ground, 
having  the  intercepted  portion  of  the  main  line  as  base,  and  the 
vertex  near  the  obstacle.     The  deflections  at  the  points  where 
the  lines  leave  and  join  the  main  line  are  similar  and  equal,  and 


PRELIMINARY    SURVEY. 


19 


the  deflection  at  the  vertex  is  double  in  amount  and  opposite 
in  direction.  No  calculation  is  necessary,  for  the  error  in 
measurement  due  to  the  deviation  is  too  small  to  be  noticed, 
and  since  the  main  line  is  immediately  resumed,  the  calculated 
bearings  of  the  auxiliary  lines  are  unnecessary.  Should  the 
point  where  the  second  line  joins  the  main  line  prove  irnsuit- 
able  for  a  transit  point,  the  second  line  may  be  produced  to 
any  convenient  point  beyond,  and  so  go  to  form  an  isosceles 
triangle  on  the  opposite  side  of  the  main  line,  the  triangle 
being  completed  by  running  a  third  line  parallel  to  the  first, 
and  equal  to  the  difference  of  the  first  and  second.  Again, 
the  second  line  may  encounter  a  serious  obstacle  before  reach 
ing  the  main  line.  To  avoid  this  a  parallel  to  the  main  line 
may  be  run  from  the  end  of  the  first  line  for  a  con- 
venient distance,  and  there  the  second  line  be  put  in 
parallel  and  equal  to  its  first  position,  as  before  de- 
scribed, thus  forming  a  trapezoid. 

47.  The  following  general  solution  of  this 
problem  allows  the  engineer  to  make  use  of  any 
number  of  auxiliary  lines,  provided  that  none  of 
them  make  an  angle  of  much  more  than  one  degree 
with  the  main  line,  with  a  certainty  of  resuming  the 
main  line  in  position  and  direction  at  the  extremity 
of  any  course  desired,  and  without  necessitating 
any  trigonometrical  calculation.  It  is  based  on  the 
assumption,  practically  true  for  small  angles,  that 
the  sines  are  proportional  to  their  angles,  and  is  ex- 
pressed by  the  following  rule  : 

Call  all  deflections  to  the  right  plus,  and  all  to 
the  left  minus;  multiply  the  length  of  each  course 
in  feet  by  the  algebraic  sum  in  minutes  of  all  the 
auxiliary  deflections  made  to  reach  that  course; 
take  the  algebraic  sum  of  these  products,  and 
when  the  sum  equals  zero  the  extremity  of  the  last 
course  will  be  on  the  main  line.  The  deflection 
required  at  that  point  to  give  the  direction  of  the  main  line 
is  equal  to  the  algebraic  sum  of  all  the  preceding  deflections, 
but  taken  with  the  contrary  sign. 

Thus,  if  we  have  left  the  mam  line  at  A,  and  run  by  these 
notes:     (Fig.  l.> 


FIG.  1. 


20 

Sta-  Defl.  Dist.                      Factors.                  Products. 

A  16'  R  190  g>  «  +  16  X  190  =  -f  3040 

B  31'  L  120  |  |  -  15  X  120  =                -  1800 

C  18'  R  175  1  8  +    3  X  175  =  -f    525 

I)  13'  L  265  f  |  -  10  X  265  =               -  2650 

E  »    15'  R  *  |  -f    5  X    (?) 

3565  -  4450 
and  their  algebraic  sum  is  —    885 

Therefore  to  render  the  sum  zero  we  must  add  885  as  the  pro- 
duct of  the  last  course.  But  5'  is  already  given  as  one  factor, 

885 

so  that  the  other  factor  must  be  -— -  =  177,  which  is  the  length 

o 

of  the  last  course,  giving  some  point  F  on  the  main  line.  The 
deflection  at  F  from  the  last  course  to  give  the  direction  of 
the  main  line  is 

16  —  31  +  18  -  13  +  15  =-5' 

and  changing  the  sign  we  have  —  5' ;  that  is,  the  deflection  is 
to  the  left. 

The  distance  on  the  main  line  from  A  to  F  equals  the  sum 
of  the  courses,  or  927  feet,  but  this  we  have  by  the  stations, 
which  have  been  kept  by  stakes  in  the  ordinary  way.  All  the 
stakes  on  the  auxiliary  lines  will  be  more  or  less  off  the  main 
line,  but  as  these  offsets  are  usually  very  small,  they  are  con- 
sidered of  no  consequence  on  a  preliminary  survey  through  a 
forest.  In  Fig.  1  the  offsets  are  very  much  magnified.  The 
field  notes  of  such  auxiliary  courses  should  be  kept,  not  as 
regular  notes,  but  on  the  margin  or  opposite  page,  and  in  such 
a  way  that,  while  the  line  may  be  retraced  by  them  on  the 
ground,  the  draughtsman  may  see  that  it  is  not  necessary  to  plot 
them,  when  a  straight  line  ruled  and  measured  through  is  suf- 
ficient. It  is  obvious  that  in  selecting  a  closing  course  either 
the  deflection  may  be  assumed  and  the  length  calculated,  or 
vice  versa  ;  but  care  should  be  taken  to  assume  such  values  as 
do  not  involve  a  fraction  in  either  factor,  if  possible. 

48.  The  method  of  passing  an  obstacle  on  the  line  by 
auxiliary  lines  at  a  large  angle  with  the  main  line  will 
only  be  resorted  to  when  circumstances  are  such  that  the  other 
methods  mentioned  cannot  be  employed,  as  in  passing  a  build- 
ing, pond,  or  densely  wooded  swamp.  In  such  a  case  we  may 


PRELIMINARY    SURVEY.  21 

turn  a  right  angle  with  the  transit,  and  measure  accurately  one 
offset,  putting  a  transit  point  at  its  extremity,  where  another 
right  angle  will  give  a  parallel  line.  If  the  offset  prove  too  short 
for  an  accurate  backsight,  a  temporary  point  at  a  sufficient 
distance  may  be  established  for  that  purpose  on  the  offset  line 
produced  before  the  instrument  is  removed  from  the  main 
line.  If  any  other  angle  than  90°  is  used  it  should  be  selected, 
when  circumstances  permit,  so  that  the  distances  on  the  inter- 
cepted part  of  the  main  line  may  be  in  some  simple  ratio  to 
the  distances  measured  on  the  auxiliary  line.  Thus  a  deflection 
of  60°  gives  a  distance  on  the  main  line  equal  to  half  the 
length  of  the  auxiliary  course,  that  is,  . 

60°          gives  a  ratio  of  i  =  0.5 
53°  08'       "      "  "        0.6  nearly 

45°  34V     "      "  "        0.7      " 

36°  52"      "      "  "        0.8      " 

25°  50V     "      "  "        0.9      " 

the  angles  being  taken  to  the  nearest  half  minute. 

49.  If  it  be  desired  that  the  stakes  on  the  auxiliary  line 
should  stand  on  perpendiculars  through  the  true  stations 
on  the  main  line,  a  certain  correction  must  be  added  to  each 
chain  length  depending  on  the  angle  which  the  auxiliary 
makes  with  the  main  line.  If  there  is  a  fraction  of  the  chain 
at  either  end  of  the  course,  a  proportional  addition  must  be 
made  for  this.  Thus,  by  referring  to  the  table  of  external 
secants,  we  find  that  we  must  add  a  correction  as  follows: 


2°  33V- 

.  .0.1  ft.  per  chain. 

6°  45V. 

..0.7ft.  per  chain. 

3°  37'  i 

..0.2" 

7°  13V- 

..0.8" 

4°  26'  . 

..0.3" 

7°  39V- 

..0.9" 

5°  07'  . 

..0.4" 

8°  04'  . 

..1.0" 

5°  43'  . 

..0.5" 

9°  52'  . 

..1.5"   "   . 

6°  15V. 

.0.6" 

11°  22'  . 

..2.0" 

These  methods  of  suiting  the  angle  to  an  even  measure  are 
much  superior  to  assuming  an  even  number  of  degrees  deflec- 
tion, and  then  calculating  the  distance  by  trigonometry.  The 
last  table,  which  may  be  extended  indefinitely  by  reference  to 
the  table  of  Ex.  secants,  is  perfectly  adapted  to  chaining*  by 
surface  measure  on  regular  slopes  when  the  slope  angle  is 


22  FIELD 

known,  the  chain  being  lengthened  by  the  correction  corre- 
sponding to  the  slope  angle. 

50.  If  the  chain  is  lengthened  as  per  above  table  on  auxil- 
iary lines,  the  numbering  of  the  stakes  goes  on  as  usual,  but 
they  should  have  an  additional  mark  as  X  to  show  that  they 
are -off  the  main  line;  and  they  may  stand  facing  the  true 
stations  which  they  represent,  and  the  length  of  offset,  if 
known,  may  also  be  recorded  on  them.     The  leveller  will  then 
understand  that  he  is  to  read  the  rod  not  only  at  the  stakes  as 
they  stand,  but  also  at  the  true  stations,  as  nearly  as  may  be. 
The  assistant  engineer  will  always  make  a  diagram  in  his 
field  book,  showing  exactly  the  method  pursued  in  reference  to 
auxiliary  lines.     Having  passed  the  obstacle,  it  is  advisable 
to  return  to  the  main  line  by  a  course  equal  in  length  to  the 
first  auxiliary,  and  making  an  equal  angle  with  the  main  line. 
If  this  cannot  be  done  from  the  end  of  the  first  course,   a 
parallel  to  the  main  line  may  be  run  any  convenient  distance, 
and  the  return  line  then  put  in,  forming  a  trapezoid. 

51.  When  there  is  no  obstruction  to  sight  on  the  main 
line,  but  only  to  measurement,  a  transit  point  should  be 
set  in  line  beyond  the  obstacle  before  the  transit  leaves  the 
main  line,  as  St  check  on  the  other  operations,  and  the  main 
line  should  be  afterward  produced  from  this  point  by  back- 
sight on  the  main  line,  rather  than  by  deflection  from  an 
auxiliary  line. 

52.  The  main  line  should  always  be  resumed  as  soon  as 
practicable,  making  the  auxiliary  lines  the  mere  exception. 
When  a  number  of  courses  at  a  large  angle  are  likely  to  be 
required  before  the  main  line  can  again  be  reached,  it  may  be 
better  to  consider  these  as  regular  courses  of  the  survey,  and 
to  note  them  as  such.     The  simplest  method  is  always  the  best, 
because  least  likely  to  involve  mistakes. 

53.  When  the  natural  obstacles  are  so  numerous 
and  of  such  magnitude  as  to  render  any  continuous  line  of  sur- 
vey or  location  extremely  difficult,  if  not  impossible,  as  in  the 
case  of  a  bold  rocky  shore,  all  the  data  necessary  to  a  location 
should  be  gathered  with  precision  on  the  preliminary  survey, 
the  measurements  and  angles  being  taken  with  the  greatest  care, 
and  as  many  checks  as  possible  should  be  introduced  to  verify 
the  work.   In  meandering  such  a  shore  it  is  probable  that  a  large 
number  of  short  courses  will  be  used,  which  may  be  measured 


PRELIMINARY    SURVEY.  23 

correctly,  but  there  is  liability  to  error  in  the  angles.  To 
verify  the  latter  the  more  conspicuous  transit  stations  on 
prominent  points  of  the  shore  are  selected,  and  these  being 
named  by  the  letters  of  the  alphabet,  the  deflections  between 
them  are  taken  by  careful  observations  repeated  .a  number  of 
times,  as  for  a  triaugulation.  These  points,  joined  by  tie- 
lines,  then  form  a  survey  of  themselves,  much  simpler  than 
the  full  traverse.  To  obtain  the  length  of  these  tie-lines',  the 
angles  between  them  and  the  courses  meeting  at  the  same 
station  are  measured.  Then  since  each  tie-line  forms  the 
closing  side  of  a  field,  in  which  all  the  bearings  are  known, 
and  all  the  distances,  save  one,  that  one  may  be  calculated  by 
latitude  and  departures.  But  the  angles  should  first  be  tested 
for  error  in  each  complete  field,  and  if  the  error  be  large  the 
angles  must  all  Be  remeasured  until  the  error  is  found  and  cor- 
rected, but  if  very  small  it  may  be  distributed  among  the 
angles,  or  among  those  most  probably  inaccurate.  Before  cal- 
culating the  traverse  of  any  of  these  fields,  it  will  be  advanta- 
geous to  assume,  for  an  artificial  meridian,  a  line  parallel  to 
the  average  direction  of  the  shore  for  several  miles,  and  to 
refer  all  courses  to  this  meridian  for  their  bearing.  This 
meridian  is  called  the  axis  of  the  survey,  and  all  bearings 
referred  to  it  are  called  axial  bearings,  as  distinguished  from 
magnetic  bearings.  The  magnetic  bearing  of  the  axis  should 
be  some  exact  number  of  degrees,  so  as  to  facilitate  the  reduc- 
tion from  one  system  to  the  other. 

54.  In  plotting  the  map,  the  axis  is  first  laid  down,  and  then 
the  lettered  stations  in  their  respective  positions,  after  which 
the  meandering  surveys  can  be  filled  in.  The  map  being 
drawn  on  a  scale  of  one  hundred  feet  to  an  inch,  and  the  con- 
tours constructed  from  the  notes  of  the  level  and  cross-level 
parties,  the  engineer  may  project  the  location  upon  it  with 
great  certainty  and  economy  of  result.  But  he  should  calcu- 
late the  traverse  of  the  location  as  projected,  and  compare  it 
with  the  traverse  of  the  preliminary,  to  eliminate,  all  errors  in 
drafting,  before  taking  his  notes  to  the  field  to  reproduce  the 
location  on  the  ground.  Any  point  where  the  location  crosses 
the  preliminary  should  have  the  same  latitude  and  longitude 
by  the  traverse  of  either  line.  This  system,  though  laborious, 
is  the  only  one  that  will  ensure  a  successful  location  under  the 
circumstances  supposed.  Advantage  may  sometimes  be  taken 


24  FIELD   ENGINEERING. 

of  cold  weather  to  cross  bays  and  inlets  on  the  ice,  but  there 
is  great  liability  to  error  in  angles  taken  upon  the  ice,  due  both 
to  its  motion  and  to  the  sinking  of  the  feet  of  the  tripod  into 
the  ice  as  soon  as  exposed  to  the  rays  of  the  sun. 


CHAPTER  III. 
THEORY  OF  MAXIMUM  ECONOMY  IN  GRADES  AND  CURVES. 

55.  Before  commencing  the  field  work  of  location  it  de- 
volves upon  the  engineer  to  decide  as  to  which  of  the  surveyed 
routes  shall  be  adopted  as  being  most  advantageous  in  all 
respects,  and  also  to  establish  the  maximum  grade  in  each 
direction  and  the  minimum  radius  of  curve  on  that  route. 

The  general  considerations  which  guide  the  engineer  in  the 
selection  of  one  of  several  routes  for  location  are  such  as  were 
hinted  at  in  the  chapter  on  reconnoissance,  but  upon  the  com- 
pletion of  the  preliminary  surveys  he  has  at  hand  a  large 
amount  of  information  which  enables  him  to  consider  this 
important  question  much  more  in  detail.  Unless  his  instruc- 
tions are  explicitly  to  the  contrary,  he  may  assume  it  to  be  his 
duty  to  find  the  best  line,  or  that  one  which,  for  a  series  of 
years  following  the  completion  of  the  road,  will  require  the 
least  annual  expense,  including  interest  on  first  cost.  The 
finances  of  the  company  may  be  so  limited  as  not  to  permit 
the  construction  of  the  best  line  at  once,  and  it  may  then  be 
the  duty  of  the  engineer  to  select  the  cheapest  line,  or  that  of 
least  first  cost,  as  a  temporary  expedient,  with  the  expectation 
of  building  the  road  at  its  best  when  the  improved  credit  of 
the  company  will  permit.  But  generally  he  will  be  able  to 
build  the  cheaper  portions  of  the  best  line  at  once,  only  making 
deviations  and  introducing  heavier  grades  at  the  expensive 
points  to  avoid  a  cost  beyond  the  present  means  at  his  com- 
mand. The  selection  of  the  best  line  may  be  a  question  as 
between  different  routes  or  as  between  different  grades  and 
curves  on  the  same  route.  We  will  consider  the  latter  case 
first. 

56.  To  solve  the  problem  of  true  economy  we  must 
determine  the  actual  expense  both  of  building  and  operating 


MAXIMUM   ECONOMY   IK   GBADES,  ETC.  25 

the  line  at  a  given  maximum  grade,  and  also  what  changes  will 
be  made  in  these  expenses  by  a  change  in  that  maximum.  We 
have  then,  on  one  hand,  the  annual  interest  upon  the  original 
cost,  and,  on  the  Other,  the  annual  expense  of  operating  the  road. 
The  best  grade  is  that  which  will  render  the  sum  of  these  two  a 
minimum.  Both  forms  of  expense  consist  of  two  parts:  one 
that  is  affected  by  a  change  in  grade,  and  the  other  that  is  not. 
Clearly  the  former  is  the  only  one  we  have  to  consider  in  either, 
since  when  the  sum  of  the  variable  portions  is  a  minimum,  the 
sum  total  will  be  a  minimum  also.  The  varying  portions  then 
are  functions  of  the  grade,  though  independent  of  each  other. 
If,  therefore,  we  let  z1  represent  the  maximum  grade  in  feet 
per  mile,  and  let  x  represent  the  corresponding  value  of  that 
portion  of  the  annual  expense  which  varies  with  the  grade, 
and  establish  the  relation  existing  between  the  two,  we  shall 
have  x  =f(z').  Similarly  if  we  let  y  represent  the  interest  on 
so  much  of  the  first  cost  as  is  affected  by  grade,  we  shall  have 
y=f  (z'}.  The  problem  then  is  to  find  that  value  of  z'  which 
shall  render 

x  -j-  y  =  a  minimum. 

Let  us  now  seek  the  complete  expression  represented  by 
x=f(z').' 

The  elements  that  enter  into  this  expression  are  numerous, 
and  will  be  considered  in  succession. 

5  7 .  The  traction  of  an  engine  is  the  force  with  which 
it  pulls  a  train,  and  is  limited  by  the  reaction  of  the  drivers 
against  the  rails.  It  depends  on  the  weight  upon  each  driver,  the 
number  of  drivers,  and  the  coefficient  of  friction.  The  weight 
on  one  driver  should  not  exceed  12,000  Ibs.,  and  is  usually  less 
than  this.  If  the  exact  proportions  of  engine  that  will  be 
used  on  the  road  are  not  known,  the  weight  per  driver  may 
be  assumed  at  10,000  Ibs.,  with  4  drivers  for  ordinary  grades 
and  traffic,  or  at  11,000  Ibs.  with  6  drivers,  if  the  grades  are 
steep  and  the  tonnage  large.  For  extraordinary  grades  special 
engines  are  required,  having  8  or  10  drivers.  Tlie  coefficient 
of  friction,  called  also  the  adhesion,  varies  from  .09  to  .37, 
these  being  the  extremes  on  record.  The  lowest  is  due  to 
extremely  unfavorable  circumstances,  as  sleet  and  frost;  the 
highest  doubtless  to  the  use  of  sand,  though  not  so  stated  in 
the  record.  The  more  common  range  of  values  is  from  .15  to 


FIELD 


.25.  For  our  present  purpose  it  will  be  assumed  at  .20,  so 
that  if  a  4-driver  engine  has  10,000  Ibs.  on  each  driver,  its 
traction  is  40,000  X  .20  =  8000  Ibs.  when  hauling  its  maximum 
train. 

58.  The  expense  of  running  an  engine  one  mile,  hauling 
a  train,  on  the  proposed  road,  can  only  be  estimated  from  the 
experience  on  other  roads  similarly  situated.  The  expense  is 
composed  of  the  items  of  fuel,  water,  oil  and  waste,  repairs 
(including  renewals),  wages  of  conductor,  engineer,  and  fire- 
roan,  engine-house  expenses,  and  interest  on  first  cost  of 
engine  and  engine-stall.  The  range  and  approximate  average 
of  these  items  is  here  given : 


ITEMS. 

4-DuivER  ENGINE.     4-DuiVER 

6-DRiVER 

8-DRIVER 

Lowest. 

Highest. 

Average. 

Average. 

Average. 

Fuel  

$0.050 
.001 
.004 
.050 
050 
025 
.0^5 

$0.210 
.010 
.030 
.150 
.100 
060 
.038 

$0.100 
.004 
.006 
.080 
.075 
.035 
,030 

$0.165 
008 
.008 
.104 
.075 
.050 
.038 

$0.213 
.008 
.010 
.133 
.075 
.060 
.047 

Water  
Oil  and  waste  
Repairs  and  renewals 
Wages 

Engine-house  .  
Interest 

Totals  

.205 

.598 

.330 

.446 

.546 

In  a  given  case  the  probable  value  of  each  item  should  be 
estimated  separately,  and  the  sum  taken  afterwards.  In  the 
above  averages  each  engine  is  supposed  to  haul  its  maximum 
train.  The  relative  expense  of  the  several  classes  of  engines 
has  not  been  established  conclusively. 

59.  The  resistance  offered  to  the  motion  of  a  railway 
train  is  occasioned  by  a  variety  of  causes,  concerning  which 
a  great  deal  of  -uncertainty  exists  as  to  their  relative  effect. 
An  investigation  which  should  seek  to  determine  the  exact 
amount  of  each  partial  resistance,  and  then  by  a  summation 
derive  the  total,  would  be  tedious,  and,  in  the  present  state  of 
our  knowledge,  unsatisfactory.  We  shall  therefore  simply 
group  the  resistances  under  three  general  heads,  namely: 

Resistance  due  to  uniform  motion  on  a  straight,  level  track ; 

Resistance  due  to  grade ; 

Resistance  due  to  curvature. 


MAXIMUM   ECONOMY   IN   GRADES,  ETC.  '27 

6O.  The  first  of  these,  considered  as  an  aggregate  of 
the  various  items  of  friction  in  engine  and  train,  of  oscillations 
and  impacts,  and  of  resistance  of  the  atmosphere,  is  found  to 
vary  nearly  or  quite  as  the  square  of  the  velocity.  The  fric- 
tion of  an  engine  is  greater  in  proportion  to  its  weight  than 
that  of  a  car,  owing  to  its  many  moving  parts,  so  that  the 
resistance  of  a  short  train  is  greater  in  proportion  to  its  total 
weight  than  that  of  a  long  train.  The  resistance  of  the  atmos- 
phere is  greater  also  in  proportion  to  the  weight  of  a  short 
train  than  of  a  long  one.  An  empty  train  will  offer  more 
resistance  in  proportion  to  its  weight  than  a  loaded  one.  A 
formula  which  shall  express  the  resistance  of  a  train  to  uni- 
form motion  must  include  at  least  the  velocity  and  the  weight 
of  the  train  and  engine. 

The  following  empirical  formula  is  based  upon  a  careful 
investigation  of  all  such  records  of  experiments  on  the  subject, 
several  hundred  in  number,  as  have  come  to  the  author's  notice, 
and  is  believed  to  give  results  agreeing  closely  with  the  average 
experience  and  practice  of  tlie  present  day.  It  is  designed  to 
give  the  resistance  per  ton  for  all  trams,  whether  freight  or 
passenger,  and  at  any  velocity,  under  ordinary  circumstances. 
Accidental  circumstances,  such  as  the  state  of  the  weather, 
and  the  condition  of  the  road-bed,  rails,  and  rolling  stock,  may 
largely  modify  the  resistance,  but  these,  of  course,  are  not 
taken  account  of  in  the  formula. 

Let  V  =  velocity  of  train  in  miles  per  hour, 
"    E  =  weight  of  engine  and  tender  in  tons, 
"   W  =  weight  of  cars  in  tons, 
"    T  —  weight  of  freight  in  tons, 
"    q  —  resistance  to  uniform  motion  in  Ibs.  per  ton. 

We  then  have  the  formula 


=  5.4        .006 


61.  The  second  resistance  considered  is  that  due  to 
gravity  in  grades.  It  varies  in  the  exact  ratio  of  the  rise  to  the 
length  of  the  grade. 

Let  (?8    =  rise  of  grade  in  feet  per  station. 
"    Gm  =  rise  of  grade  in  feet  per  mile. 
"    q      =  resistance  in  pounds  per  ton  due  to  grade. 


28  FIELD 

Then, 

tf  =  2240        =  23.4.0. 


(2) 


62.  The  third  resistance  considered  is  that  due  to 
curvature  of  the  track.  This  resistance  is  due  to  the  friction 
of  the  wheels  upon  the  top  of  the  rail,  and  of  their  flanges  upon 
the  side  of  the  rail.  The  top  friction  is  lateral,  due  to  the 
oblique  position  of  the  wheel  on  the  rail,  and  longitudinal,  due 
to  the  greater  length  of  the  outer  rail,  since  both  wheels  are 
rigidly  attached  to  the  axle.  The  flange  friction  is  due  to  the 
reaction  of  the  top  friction,  which,  combined  with  the  parallel- 
ism of  the  axles,  throws  the  truck  into  an  oblique  position  on 
the  track.  A  forward  flange  presses  the  outer  rail,  while  a  rear 
flange  is  usually  in  contact  with  the  inner  mil.  The  centri- 
fugal force  of  the  car  will  increase  the  pressure  on  the  outer 
rail,  unless  the  ties  are  inclined  at  an  angle  sufficient  to  coun- 
terbalance this  force.  But  if  the  ties  are  inclined  too  much, 
or  the  velocity  is  less,  the  pressure  on  the  inner  rail  will  be 
increased.  An  uneven  track  will  cause  the  truck  to  pursue  a 
zigzag  course,  increasing  the  resistance  considerably. 

Experiments  for  determining  the  amount  of  curve  resistance 
have  been  neither  numerous  nor  very  satisfactory,  but  the 
generally  accepted  conclusion  is  that  the  resistance  is  a  little 
less  than  half  a  pound  per  ton  on  a  one-degree  curve,  and  that 
it  varies  as  the  degree  of  curve.  On  European  roads,  how- 
ever, it  is  estimated  at  about  one  pound  per  ton  per  degree  of 
curve,  owing  largely  to  the  form  of  rolling  stock  used. 

63.  Let   g"  =  curve  resistance  in  pounds  per  ton  on  any 

curve, 
and  D   =  degree  of  curve. 

Then,  assuming  the  resistance  per  ton  on  a  one-degree  curve 
at  0.448,  we  have  for  any  other  curve 

q"  =  0.448Z>  (3) 

To  ascertain  what  grade  upon  a  straight  line  will  offer  the 
same  resistance  as  a  given  curve;  substitute  the  value  of  q" 
for  q'  in  eq.  (2)  and  solve  for  G;  whence 

&,  =  .ow  i  4 

#„,  =  1.056,0  \ 


MAXIMUM   ECONOMY   IN   GRADES,  ETC.  29 

For  definition  of  degree  of  curve,  see  Art.  84. 

O4.  It  is  evident  that  grades  and  curves,  by  their  resistances, 
fix  a  limit  to  the  weight  of  a  train  which  a  given  engine  can 
haul  over  them. 

A  locomotive  is  usually  built  with  such  a  surplus  of  boiler 
and  cylinder  capacity  that  its  power,  at  ordinary  velocities,  is 
limited  by  the  adhesion  of  the  drivers,  so  that  the  adhesion  is 
the  proper  measure  of  the  tractive  force. 

To  find  an  expression  for  the  maximum  train  which  a  given 
engine  can  haul  over  a  given  grade  and  curve: 

Let    P  =  tractive  force  of  engine  in  pounds, 
"      T'  =  weight  of   paying  load  in   tons  per  maximum 

train, 
"     W  =  weight  in  tons  of  cars  carrying  the  load  T'. 

Then  for  uniform  motion,  at  a  given  velocity. 

Let  t  =  average  load  of  one  car  in  tons 
"  w  =  average  weight  of  one  car  and  load  in  tons. 

Then  W  +  T'  =  y T',  substituting  which  in  eq.  (5)  we  derive 

r^f-r^-p^-tf)  (6) 

In  this  equation  q  =  the  resistance  per  ton  due  to  uniform 
motion,  q'  =  the  resistance  per  ton  due  to  the  maximum  grade 
opposed  to  the  direction  of  the  train,  and  q"  =  the  resistance 
per  ton  due  to  the  sharpest  curve  on  that  grade. 

For  accelerated  motion  the  reaction  of  inertia  of  the  train 
must  be  added  to  the  above  resistances.  This  is  estimated  at 
^q,  in  order  that  a  train  starting  from  rest  may  acquire  the 
requisite  maximum  velocity,  even  on  a  maximum  grade,  in  a 
reasonable  time,  say  from  3  to  6  minutes.  Therefore,  for 
accelerated  motion, 


Now,  the  values  of  T  and  q  involve  each  other,  but  if  we 
accent  W  and  T  in  eq.  (1)  the  value  of  q  becomes  that  used  in 


30  FIELD   ENGINEERING. 

eq.  (7),  and  we  may  eliminate  q  between  these  equations,  and 
derive  the  value  of  T'  ;  whence 

-(p-.ooo9^2F*) 

T-  Lv 

~  4  +  <l"  +  8-1  +  -009  V'2       w 

Also,  for  the  weight  of  maximum  train  and  load, 

,,         P-  .000110*  r«  (9 

~  E 


which  is  the  expression  required. 

When  there  is  no  curve  on  the  maximum  grade,  q"  is  zero; 
and  when  there  is  no  grade,  q'  is  zero;  hence  for  a  straight  level 
track  eq.  (7)  becomes 


(10) 


and  eq.  (8) 

'(P-  I 
r.__™(P      -'  &m 

8.1  +  .009  F*  to 


65.  An  engine-stage  is  a  division  of  the  road  to  which 
an  engine  is  limited,  and  over  which  it  regularly  hauls  a  train. 
Its  length  varies,  on  existing  roads,  from  50  to  200  miles  or 
more,  depending  on  the  grades,  on  the  length  of  the  whole 
line,  and  on  the  distance  between  points  favorable  for  the  loca- 
tion of  shops,  etc.  The  average  engine-stage  on  American 
roads  is  not  far  from  75  miles.  If  there  are  to  be  several 
engine-stages  on  the  proposed  line,  the  problem  of  maximum 
economy  of  grade  must  be  solved  with  reference  to  each  of 
them  separately. 

Let  L  —  length  of  engine-stage  in  miles, 
"    e  —  expense  per  engine-mile  in  dollars, 

'  "  A  =  average  annual  paying  freight  in  tons  moving  in 

one  direction,  and 

"  a  =  average  annual  paying  freight  in  tons,  moving  in 
the  opposite  direction;  and  if  these  are  not  equal,  let  A  be 
greater  than  a.  Now  T'  eq.  (8)  is  the  maximum  train-load 
which,  at  a  velocity  F,  should  be  hauled  up  steepest  grade  z' ', 

opposed  to  the  direction  of  the  tonnage  A ;  hence    ^  =  the 


MAXIMUM   ECONOMY   IN   GRADES,  ETC.  31 

number  of  trains  per  annum;  and  since  each  train  must  go 

2  T  A 

and  return,  .'.  —  «?7"  =  the  total  train-mileage  per  annum. 

It'  there  were  no  return  tonnage,  the  annual  expense  charge- 

2A.L& 
able  to  A  would  be  —  ™-,-,  but  since  some  of  the  cars  return 

loaded  with  the  freight  a,  these  are  not  chargeable  to  A,  and 
must  be  deducted  from  the  above  expression.  Hence  if  we 
denote  the  annual  expense  of  engine-mileage  by  x, 


in  which  the  value  of  the  maximum  grade  z'  is  involved  in 
the  value  of  T'. 

But  we  may  obtain  an  expression  for  x  in  terms  of  z'  ;  for, 
at  any  given  velocity,  the  resistance,  q0,  on  a  level  is  equal  to 
the  resistance  due  to  a  certain  grade  z0,  the  value  of  which  is, 
by  eq.  (2),  for  uniform  motion, 

33 

/y       -    _  /y 

0  ~  14  q° 
So  the  resistance,  q,   to  motion  up  a  grade  z'   is  equal  to 

qq 

the  resistance  due  to  some  grade  z  —  —  q,  the  total  resistance 

being  that  due  to  the  combined  grades  z  -f-  z'.  Now,  since 
the  gross  weight  of  a  maximum  train,  under  a  constant  engine 
power,  is  inversely  as  the  resistances,  we  have,  for  conditions 
of  accelerated  motion  : 


whence 

T'  =  -  (12) 

in  which  T'0  —  maximum  train-load  on  a  level  line.     Substi- 
tuting this  value  of  T'  in  eq.  (11)  we  have 

x  =  —       *z  +  Z' (2A  -  a)  Le      (13) 

w 
which  is  the  complete  expression  for  x  =  f  (z")  required. 


32  FIELD 


66.  Could  we  also  find  a  complete  expression  for  y  —f  (2'}, 
we  might  then  proceed  to  find,  by  analysis,  that  value  of 
z1  which  would  render  x  -j-  y  —  a  minimum.  But  the  value 
of  y  cannot  be  formulated,  since  it  depends  on  the  accidental 
features  of  the  country  through  which  the  line  passes;  it  can 
only  be  determined  for  any  given  value  of  z'  by  an  estimate 
based  on  the  survey.  We  therefore  resort  to  a  graphical 
solution. 

Equation  (13)  is  the  equation  of  a  curve  in  the  plane  ZX, 
Fig.  2.  If  we  assume  several  values  of  z',  and  calculate  the 
corresponding  values  of  x,  we  may  lay  these  off  by  scale  on 
the  axes  of  Z  and  X  respectively,  and  so  obtain  several  points 


& 


'Y  /     O  x" 

FIG.  2. 

through  which  the  curve  of  annual  expense  may  be  drawn. 
We  then  make  estimates  of  the  cost  of  constructing  the  road 
at  the  same  values  of  z',  and  taking  the  annual  interest  of 
each  estimate  as  an  ordinate  y  to  OZ  in  the  plane  ZT,  we  lay 
it  off  to  scale  at  the  proper  height,  thus  obtaining  a  series  of 
points  in  the  plane  ZT,  through  which  the  curve  of  annual 
interest  on  first  cost  may  be  drawn.  If  now  we  suppose  the 
plane  ZYto  be  revolved  to  the  left  about  the  axis  OZ  until 
it  coincides  with  the  plane  OX,  as  in  Fig.  2,  we  shall  see 
that  the  two  curves  are  convex  to  OZ  and  to  each  other.  The 
shortest  horizontal  line  intercepted  by  them  indicates  the 
minimum  value  of  (x  -j-  y),  and  the  point  where  this  line  cuts 
the  axis  OZ  indicates  the  corresponding  value  of  z',  which  is 
the  one  required.  If  tangents  be  drawn  to  the  curves  at  the 
points  where  the  shortest  horizontal  line  intersects  them,  the 
tangents  will  be  parallel  to  each  other.  Any  convenient  scales 
may  be  used  to  lay  off  the  values  of  z'  and  x,  provided  that 
the  values  of  x  and  y  be  laid  off  to  the  same  scale.  It  is  well 


MAXIMUM   ECONOMY   IN   GKADES,  ETC.  33 

to  reduce  all  the  values  of  x  by  an  amount  common  to  them 
all,  and  the  same  with  respect  to  values  of  y,  before  laying 
them  off  to  scale.  This  will  bring  the  two  curves  nearer 
together  without  altering  their  form. 

67.  To  facilitate  the  calculation  of  x,  we  give  on  the  next 

page  a  table  of  values  of  -^  for  several  engines,  using  eq.  (8) 

for  this  purpose.  The  value  of  x  is  therefore  found,  eq.  (11) 
or  (13),  by  multiplying  (2A  —  a)  Le  by  the  proper  tabular 
number,  under  conditions  assumed  as  follows: 

t     =10  tons  of  freight  per  car-load; 

w   —  18  tons  per  car  and  load; 

F  =  12  miles  per  hour. 

Fora    4-driver  engine,  E  —  42     tons;  P  =    8100  Ibs. 

Fora    6-driver  engine,  E=  49.5     "      P  =  12600   " 

For  an  8-driver  engine,  E  =  59.4     "      P=  17280    " 

Substituting  these  values  in  eq.  (8),  and  making  q"  =  0, 
we  find  the  maximum  loads  of  freight  which  the  several 
engines  can  haul  up  the  grade  whose  resistance  is  q'.  The 
reciprocals  of  these  loads  are  given  in  the  table  opposite  the 
grades  noted  in  the  first  and  last  columns. 

68.  Since  q"   is  made   zero,  the   grades  in  the  table  are 
assumed  to  be  on  straight  lines.     In   locating  a  road,  the 
maximum  grade  should  be  reduced  on  a  curve  by  the  amount 
of  the  equivalent-grade  of  the  curve,  eq.  (4),  so  that  the  resist- 
ance may  be  no  greater  on  the  curve  than  elsewhere.     But 
grades  less  than  the  maximum  need  not  necessarily  be  reduced 
for  the  curves  upon  them,  unless  the  sum  of  the  grade  and  the 
curve-equivalent  exceeds  the  maximum. 

69.  For  an  example,  let  us  suppose  that  a  certain  engine- 
stage  is  to  be  80  miles  long,  and  that  an  estimate  of  the  cost  of 
construction  has  been  made,  based  on  a  ruling  or  maximum 
grade  of  52.8  ft.  per  mile  against  the  heavier  traffic,  and  that 
the  annual  interest  on  the  estimate  amounts  to  $168,000. 

Let  us  further  suppose  that  the  average  traffic  u>  one  direc- 
tion is  estimated  at  375  000  tons  per  annum,  and  in  the  other 
direction  at  125  000  tons,  that  it  is  decided  to  use  6-driver 
engines,  and  that  the  expense  per  engine-mile  is  estimated  at 
40  cents ;  hence  (2 A  —  a)Le  —  20  000  000.  We  are  now  required 
to  find  the  most  economical  maximum  grade. 

We  first  select  at  least  two  other  maximum  grades,  and  having 


FIELD    E^GLNEEKING. 


TABLE  OF  RECIPROCALS  OF  T'. 


t  -  10,  W  =  18. 


<?„. 

#  =  42 
P=8100 

Diff. 

.EJ=49.5 
P=  12600 

Diff. 

E=59A 
P  =  17280 

Diff. 

z' 
ft.  per 
mile. 

4.0 
3.9 
3.8 
3.7 
3.6 
8.5 
3.4 
3.3 
3.2 
3.1 

.0479  844 
.0457  399 
.0436  036 
.0415  679 
.0396  259 
.0377712 
.0359  980 
.0343  012 
.0326  759 
.0311  176 

22445 
21  363 
20  357 
19420 
18547 
17732 
16968 
16253 
15583 

.0241  385 
.0232431 
.0223  739 
.0215297 
.0207094 
.0199  120 
1  .0191367 
1  .018:3824 
.0176483 
.0169336 

8954 
8692 
S442 
8203 
7974 
77,3 
7543 
7341 
7147 

.0162847 
.0157250 
.0151  786 
.0146  150 
.0141  238 
.0136  146 
.0131  168 
.0126  302 
.0121  545 
.0116892 

5597 
5464 
5336 
5212 
5092 
4978 
4866 
4757 
4653 

211.20 
205.92 
200.64 
195.36 
15)0.08 
184.80 
179.52 
174.24 
168.96 
163.68 

14  952 

6960 

4553 

3.0 
2.9 
2.8 
2.7 
2.6 
2.5 
2.4 
2.3 
2.2 
2.1 

.0296  224 
.0281  864 
.0268061 
.0254  784 
.0242005 
.0229695 
.0217828 
.0206  381 
.0195333 
.0184  663 

14360 
13803 
13277 
12779 
12310 
11867 
11  447 
11048 
10  670 

.0162  376 
.0155  596 
.0148  988 
.0142  546 
.0136  264 
.0130  136 
.0124  157 
.0118  321 
.0112622 
.0107  056 

6780 
6608 
6442 
6282 
6128 
59T9 
5836 
5699 
5566 

.0112  339 
.0107  884 
.0103  524 
.0099255 
.0095  075 
.0090  981 
.0086  970 
.0083040 
.0079  188 
.C075  413 

4455 
4360 
4269 
4180 
4094 
4011 
3  930 
3852 
3775 

158.40 
153.12 
147.84 
142.56 
187.28 
132.00 
126.72 
121.44 
116.16 
110.88 

10311 

543G 

3701 

2.0 
1.9 
.8 
.7 
.6 
.5 
.4 
.3 
.2 
1.1 

.0174  352 
.0164382 
.0154  736 
.0145  399 
.0136356 
.0127593 
.0119099 
.0110  860 
.0102  865 
.0095  104 

9970 
9646 
9337 
9043 
8763 
8494 
8239 
7995 
7761 

.0101  620 
.01)96  308 
.0091  115 
.0086  039 
.0081  074 
.0076  218 
.0071  467 
.0066818 
.0062267 
.0057810 

5312 
5193 
5076 
4965 
4856 
4751 
4649 
4551 
4457 

.C071  712 
.0068  082 
.0064522 
.0061  029 
.0057  602 
.0054239 
.0050  938 
.0047  698 
.0044  517 
.0041  393 

3630 
3560 
3  493 
3427 
3363 
3301 
3240 
3181 
3124 

105.60 
100.32 
95.04 
89.76 
84.48 
79.20 
73.92 
68.64 
63.36 
58.08 

7538 

4365 

3069 

1.0 
.9 
,8 
.7 
.6 
.5 
.4 
.3 
.2 
.1 
0.0 

.0087  566 
.0080242 
.0073  123 
.0066200 
.0059466 
.0052  913 
.0046533 
.0040320 
.0034  268 
.0028370 
.0022620 

7324 
7119 
6923 
6734 
6553 
G380 
G213 
6052 
5898 
5750 

.0053445 
.00491  VI 
.0044984 
.0040880 
.0036858 
.0032915 
.0029  050 
.0025  259 
.0021  540 
.0017  892 
.0014  312 

4274 
4187 
4104 
4022 
3943 
3865 
3791 
3719 
3648 
3580 

.0038324 
.0035309 
.0032347 
.0029  437 
.0026  577 
.0023766 
.0021  002 
.0018  284 
.0015612 
.0012  984 
.0010  399 

3015 
2962 
2910 
2860 
2811 
2764 
2718 
2672 
2628 
2585 

52.80 
47.52 
42.24 
36.96 
31.68 
26.40 
21.12 
15.84 
10.56 
5.28 
0.00 

MAXIMUM   ECONOMY   IK    GBADES,  ETC. 


35 


made  an  estimate  of  the  cost  of  constructing  the  road  upon 
each,  take  the  annual  interest  of  each,  as  in  the  first  case.  Let 
us  suppose  the  two  ruling  grades  thus  selected  to  be  73.92  ft. 
and  31.68  ft.  per  mile,  or  1.4  ft.  per  station  and  0.6  ft.  per 
station,  and  the  interest  on  the  estimates  to  be  $145  596  and 
$204  388  respectively,  giving  the  following  statement: 


<?8. 

y. 

1st  diff. 

Sddiff. 

1.4 

145  596 

22404 

1.0 

168  000 

13984 

0.6 

204  388 

36388 

Interpolating  by  second  differences,  we  have  the  complete 
statement: 


G> 

y 

diff.  y. 

diff.  x. 

X. 

*+y. 

z'. 

1.4 
1.3 
1.2 
1.1 
1.0 
0.9 
0.8 
0.7 
0.6 

145  596 
149886 
155  050 
1(51088 
168000 
175786 
184446 
193  980 
204388 

4290 
5164 
6038 
6  912 
7786 
8(160 
9534 
10408 

9298 
9102 
8914 
8730 
8548 
8374 
8208 
8044 

142934 
133636 
124534 
115620 
106890 

73716 

274890 
274128 
274414 

73.92 

52.80 
47.52 
42.24 

31.68 

The  numbers  in  the  fourth  and  fifth  columns  are  obtained 
as  follows :  the  values  assumed  above  give  us  (2A  —  a)  Le  = 
$20  000  000,  and  this  multiplied  by  the  tabular  differences  in 
the  preceding  table  for  a  6-driver  engine,  gives  the  numbers  in 
the  fourth  column.  We  now  observe  that  the  differences  of 
x  and  of  y  increase  in  opposite  directions,  therefore  at  some 
point  they  will  be  equal ;  and  a  simple  inspection  shows  us  that 
this  point  is  at  or  near  the  grade  of  0.9,  which  is  therefore 
the  grade  required.  We  now  multiply  the  tabular  number  for 
0.9,  and  a  6-driver  engine  by  $20  000  000,  for  thu  number  in 
the  fifth  column,  and  this  added  to  the  value  of  y  on  the 
same  line  gives  the  sum  of  (x  -\-  y}  for  the  most  economical 
grade.  This  of  course  is  not  the  total  annual  outlay  of  the 
road,  or  engine-stage,  because  many  items  of  expense  which 
are  independent  of  a  maximum  grade  have  not  been  con- 
sidered. 


36 


FIELD   E^GI^EEKISTG. 


If  an  8-driver  engine  were  to  be  used,  and  the  expense  per 
engine-mile  estimated  at  50  cts.,  then  (2 A  —  a)Lc  —  $25  000  000 ; 
hence 


<£ 

y. 

diff  .  y. 

diff.  x.           x. 

x-\-y. 

z'. 

1.1 

1.0 

0.9 

161088 
168000 
175786 

6912 

7786 

7673 
7538 

95810 

263  810 

5/*  .  80 

indicating  a  saving  of  $10  318  per  annum  in  the  case  supposed 
by  using  8-driver  engines,  although  on  a  steeper  ruling  grade. 
On  the  other  hand,  should  we  adopt  4-driver  engines,  and  esti- 
mate the  expense  per  engine-mile  at  30  cents,  we  should  find 
the  most  economical  grade  to  be  0.7  per  station  and  (x  -f-  y) 
=  $293  280,  showing  a  loss  in  this  case  of  $19  152  per  annum, 
as  compared  with  the  results  of  6-driver  engines. 
It  should  be  remembered  that  the  table  §  67  is  prepared  on 


the  assumption  that  the  ratio  —  = 


If  cars  are  to  be  used 


giving  for  full  loads  ,ftny  other  ratio,  —  ,  ,  a  new  table  may  be 
prepared  by  multiplying  each  tabular  number  by  -—  X  -3  . 

lo         t 

The  velocity  adopted  of  12  miles  per  hour  is  sufficient  for 
ordinary  grades.  When  the  maximum  grade  is  very  low,  it 
would  be  better  to  use  15  or  18  miles  an  hour  in  calculating 
the  value  of  x. 

7O.  Since  #,  eq.  (11),  varies  directly  as  L,  it  is  important 
that  an  engine-stage  having  heavy  grades  should  be  short.  Its 
length,  however,  must  be  consistent  with  the  economical 
length  of  the  adjoining  engine-stages,  and  with  the  amount  of 
work  which  an  engine  ought  to  perform  dally.  The  most 
favorable  condition  for  a  road  would  be  that  in  which  all  the 
engine-stages  were  operated  at  equal  expense.  But  if,  to 
secure  this  result,  the  engine-stage  of  heavy  grades  must  be 
unreasonably  reduced  in  length,  it  will  be  better  to  adapt  the 
grades  to  the  use  of  two  engines  per  train. 

7  1.  The  maximum  grade  z',  opposed  to  the  heavier  tonnage 
A,  having  been  determined,  we  have  now  to  consider  what  is 
the  limit  to  grades  in  the  opposite  direction.  The  engines  are 


MAXIMUM   ECOKOMY   IN   GRADES,  ETC.  37 

supposed  to  haul  their  maximum  loads  in  moving  the  ton- 
nage A,  and  since  the  return  tonnage,  a,  is  less  than  A,  the 
engines,  in  returning,  will  not  be  worked  to  their  full  capacity 
if  they  encounter  no  grades  steeper  than  z' '.  We  therefore 
have  a  margin  of  power  in  the  returning  engines  which  may 
be  taken  advantage  of  to  cheapen  the  cost  of  construction,  or 
to  shorten  the  line,  by  introducing  grades,  steeper  than  s1 ', 
against  the  lighter  traffic. 

The  weight  of  a  maximum  train  moving  up  the  grade  z'  is, 
eq.  (9),  W  -\-  T' ;  the  weight  of  the  train  returning  will  be 


Substituting  this  in  place  of  (W  -f-  T'),  eq.  (9),  and  solving  for 
q',  we  find  the  resistance  due  to  a  maximum  grade  opposed  to 
the  returning  train.  Whence,  by  eq.  (2),  if  we  let  Z  —  the 
maximum  return  grade,  and  make  q"  =  0, 

.   _  33  flo9 


Inasmuch  as  the  value  of  Z  varies  with  every  change  made 
in  2',  the  engineer,  when  estimating  the  cost  of  construction 
upon  the  basis  of  any  maximum  grade  z',  should  take  care. 
that  the  return  grade  Z  nowhere  exceeds  its  limit  as  .given  by 
the  last  equation  (14).  In  the  example,  §69,  z'  —  47.52;  hence 
T  —  203.37,  eq.  (8).  Substituting  these  values,  in  eq.  (14),  we 
find  Z=  81.25,  which  is  therefore  the  limit  for  return  grades 
in  this  case.  With  regard  to  curves  on  the  maximum  grade, 
see  §68. 

qq 

72.  If  ineq.  (1)  we  lets  =  -TJ  q  be  the  grade  per  mile  which 

offers  a  resistance  equal  to  the  resistance  to  uniform  motion 
on  a  level,  we  have 


When  F=  20  this  becomes 

(15i) 


38  FIELD 

which  is  the  grade  down  which  a  train,  whose  weight  is  (E-\- 
W-\-  T),  if  started  at  20  miles  an  hour,  will  continue  tp  move 
at  that  speed  without  steam  or  brakes.  As  that  speed  is  not 
objectionable,  so  the  grade  z,  which  induces  it  is  not,  pro- 
vided it  does  not  exceed  the  values  of  z'  or  Z  respectively, 
determined  with  reference  to  economy.  For  the  extra  work 
done  by  the  engine  in  ascending  one  grade  z  is  utilized  in 
descending  the  next;  and  the  net  result  is  the  same  as  though 
the  two  were  replaced  by  a  uniform  grade.  The  engineer 
therefore  is  not  warranted  by  economic  considerations  in 
reducing  undulating  grades  which  do  not  exceed  z  to  a  uni- 
form grade,  when  to  do  this  would  cause  any  increase  in  the 
cost  of  construction,  unless  z  exceeds  the  grades  z'  or  Z  of 
maximum  economy. 

73.  But  when  grades  exceed  z,  eq.  (15J),  the  resulting 
speeds  of  the  maximum  train  become  too  great,  and  the  neces- 
sary application  of  the  brakes  absorbs  a  portion  of  the  power 
previously  expended  in  gaining  the  summit,   which  is  thus 
worse  than  wasted,  since  it  increases  the  wear  and  tear  of 
machinery  and  track.     Therefore  the  engineer  is  justified  in 
spending  a  certain  sum  of  money  in  reducing  grades  which 
exceed  z  to  that  limit.     A  calculation  of  the  loss  of  power  due 
to  the  use  of  brakes  on  a  grade,  and  of  the  cost  of  that  lost 
power,  together  with  the  resulting  wear  and  tear  per  annum, 
will  give  the  interest  on  the  sum  that  may  be  justifiably  spent 
in  reducing  the  grade  from  its  position  of  cheapest  construc- 
tion. 

74.  The  limit  z  is  not  constant,  but  depends  on  the  weight 
of  the  maximum  train,  which  in  turn  depends  on  z' .     It  will 
not  be  the  same  in  both  directions  unless  A  =  a,  giving  z'  =  Z. 
In  the  example  §69,  E  =  49.5  and  W  -f-  T'  =  366.07;  hence, 
eq.  (15|),  z  =  21.72  descending  in  the  direction  of  the  traffic  A. 

Also  W  +  -.  T'  =  230.49,  whence  z  —  23.34  descending   in 
A 

the  opposite  direction.     These  are  the  limits  in  this  case  at 
which  undulating  grades  cease  to  be  profitable. 

75.  We  have  finally  to  consider  the  method  for  selecting  the 
best  line  from  several  proposed  routes.     For  this  purpose  we 
determine  the  most  economical  grade  on  each  route  thought 
worthy  of  consideration,  and  calculate  the  interest  on  the 
entire  cost  of  constructing  the  line  with  that  ruling  grade,  and 


LOCATION.  39 

also  the  annual  expense  of  operating  the  line,  and  take  the  sum 
of  the  two.  That  route  is  best  in  respect  to  which  this  sum  is 
the  least. 

76.  The  value  of  saving  one  mile  in  distance  on  any  route 
is  found  by  dividing  the  sum  of  the  annual  operating  expense 
and  the  interest  on  the  cost  of  construction  by  the  rate  of 
interest,  and  the  quotient  by  the  length  of  the  line  in  miles. 

77.  We  have  now  fully  discussed  the  theory  and  developed 
the  formulae   necessary  to  the    determination   of    the    most 
economical  grades;  but  the  value  of  the  results  in  a  given 
case  depend  upon  the  correctness  of  the  engineer's  estimates 
which  enter  into  the  formulae.     These  may  seldom  prove  pre- 
cisely accurate,   yet,    if  he  can  bring  them   within  definite 
limits,  he  may  determine  the  grades  of  maximum  economy 
within  corresponding  limits.     In  the  case  of  a  finished  road 
and  in  full  operation,  however,  the  elements  of  first  cost,  of 
traffic,  and  of  operating  expenses  being  known,  an  investiga- 
tion by  means  of  the  foregoing  formulae  becomes  a  critical  test 
as  to  the  economy  of  the  location  and  grades;  and  should  the 
road  fail  to  pay  dividends,  or  be  forced  to  charge  high  rates 
of  toll,  we  can  determine,  though  perhaps  too  late,  to  what 
extent  the  location  is  chargeable  with  these  results. 


CHAPTER  IV. 
LOCATION. 

78.  A  railroad  is  said  to  be  located  when  its  centre  line  is 
established  on  the  ground  in  the  position  which  it  is  intended 
finally  to  occupy.     The  location  is  made  by  an  engineer  corps 
similar  in  its  organization  to  that  employed  on  preliminary 
surveys.     The  instruments  used  are  also  the  same,  except  that 
the  transit  is  substituted  for  the  compass,  and  usually  the  target 
rod  for  the  self-reading  rod.     The  magnetic  needle  is  never 
used  upon  the  centre  line,  except  as  a  rough  check  on  the 
transit  work.     It  is  used,  however,  to  obtain  the  direction  of 
property  lines,  roads,  and  other  topographical  data. 

79.  The  remarks  upon  transit  work  in  the  preceding 
chapter  apply  to  the  running  of  straight  lines  on  location.     All 


40  FIELD   ENGINEERING. 

field-work  on  location  should  be  done  with  accuracy  and 
fidelity.  No  guesswork,  nor  rude  approximations,  are  to  bo 
tolerated.  All  transit,  points  are  made  as  secure  and  permanent 
as  possible,  and  the  more  important  ones  are  guarded  by  other 
transit  points  set  in  safe  positions  near  by,  their  distances  and 
directions  from  the  main  point  being  recorded. 

The  stakes  for  the  stations  are  made  neatly,  and  somewhat 
uniform  in  size,  and  they  are  firmly  driven.  Sometimes  a 
small  plug  is  driven  down  flush  with  the  surface  of  the  ground 
to  indicate  the  station  point,  and  the  stake  is  then  set  near  by 
as  a  witness. 

In  locating  a  very  long  tangent  the  greatest  care  is  re- 
quired to  make  it  straight.  If  the  tangent  is  produced  from 
point  to  point  by  backsights  and  foresights,  the  observation 
should  be  repeated  in  every  instance  with  reversed  instrument, 
to  eliminate  any  possible  lack  of  adjustment,  and  to  check 
any  accidental  error.  (Indeed  it  is  proper  to  observe  this  rule 
on  curves,  as  well  as  on  tangents.)  When  some  object  in  the 
horizon  can  be  used  as  a  foresight,  it  is  preferable  to  set  the 
instrument  by  this  rather  than  by  a  backsight.  For  final  loca- 
tion, the  line  should  be  cleared  to  give  as  continuous  a  line  of 
sight  as  possible,  but  in  case  of  an  obstacle  which  cannot  be 
removed  at  the  time,  at  least  two  independent  methods  of 
passing  it  should  be  employed,  so  that  there  may  be  a  check 
upon  the  alignment  beyond. 

8O.  The  leveller  selects  his  benches  far  enough  from  the 
line  to  prevent  their  being  disturbed  during  the  construction  of 
the  road.  They  should  be  nearly  at  grade,  as  a  rule,  though  it 
is  well  to  leave  a  bench  near  a  water-course  for  reference  in  lay- 
ing out  masonry  or  trestle-work.  The  rodman  holds  the  rod 
at  every  station,  and  at  every  point  on  the  centre  line  where 
the  slope  changes  direction,  so  that  these  points  may  be  accu- 
rately defined  on  the  profile.  When  he  uses  a  target  rod,  he 
sets  the  target  as  directed  by  the  leveller,  and  after  clamping 
it,  takes  the  reading.  He  reads  to  thousandths  upon  turning 
points  and  benches,  but  only  to  tenths  of  a  foot  elsewhere,  and 
announces  the  readings  to  the  leveller  for  record.  He  also 
records  the  readings  upon  turning  points  and  benches  in  his 
own  book  as  a  check.  At  the  close  of  each  day  the  leveller 
and  rodman  compare  notes,  and  draw  a  profile  of  the  line  sur- 
veyed. (See  also  §§  28,  29,  30.) 


LOCATION.  41 

81.  The  fixing  of  the  grade-lines  upon  the  profile  is 
one  of  the  most  important  operations  connected  with  the  loca- 
tion. It  is  usually  performed  by  the  engineer  in  charge  of 
the  locating  party,  as  being  most  conversant  with  the  general 
character  and  detailed  requirements  of  the  line.  The  maxi- 
mum gradients  will  have  generally  been  determined  in  advance 
from  the  preliminary  data  by  the  principles  laid  down  in  the 
preceding  chapter,  but  the  position  of  each  grade-line,  relative 
to  the  profile  of  the  surface,  must  be  left  to  the  judgment  and 
skill  of  the  engineer.  In  general,  the  grade-line  is  so  placed 
:is  to  equalize  the  amounts  of  excavation  and  embankment, 
but  there  are  various  exceptions  to"  this  rule.  Thus,  the  exca- 
vation may  be  in  excess:  first,  when  it  is  necessary  to  pass 
under  some  other  road  or  highway,  the  grade  of  which  cannot 
be  changed;  second,  when  valuable  property  is  to  be  avoided, 
the  appropriation  of  which  would  cost  more  than  the  excava- 
tion; third,  when  the  grade  is  at  the  maximum  near  a  sum- 
mit, and  cannot  be  raised  parallel  to  itself  without  incurring 
too  great  an  expense  for  masonry,  etc.,  at  some  other  part  of 
the  line.  The  embankment  may  be  in  excess,  first,  when  the 
country  is  flat  and  wet,  in  order  to  keep  the  road-bed  well 
drained;  (the  grade-line  should  be  at  least  two  feet  above  the 
average  level  of  the  surface,  or  above  high-water  mark,  if  the 
district  is  subject  to  overflow;)  second,  in  approaching  a 
stream,  where  it  is  necessary  to  raise  the  grade  above  the 
requirements  of  navigation;  third,  when  the  cuttings  on  the 
line  are  largely  in  solid  rock,  and  a  cheaper  material  for 
embankments  may  be  conveniently  had  at  other  points; 
fourth,  in  a  district  subject  to  heavy  drifts  of  snow,  by  which 
deep  cuts  would  be  liable  to  be  obstructed;  fifth,  in  side-hill 
work,  where  there  is  danger  of  land-slips;  sixth,  when  it  is 
determined  to  supply  the  place  of  a  portion  of  an  embankment 
by  a  timber  trestle-work  or  other  viaduct. 

The  apparent  equality  of  cut  and  fill  on  the  profile  does  not 
represent  an  equality  in  fact,  owing  to  the  different  bases  and 
slopes  of  the  sections  adopted,  and  to  the  various  inclinations 
of  the  natural  surface  transversely  to  the  line.  This  is  espe- 
cially true  in  side-hill  work,  where  there  are  both  cut  and  fill 
at  every  point,  while  the  profile  shows  very  little  of  either.  In 
the  latter  case  it  is  an  excellent  plan  to  combine  with  the  pro- 
file of  the  centre  line  the  profiles  of  parallel  lines  ten 


42  FIELD   ENGINEERING. 

or  twenty  feet  either  side  of  the  centre,  and  drawn  with  differ- 
ent colored  inks,  as  these  will  indicate  tolerably  well  the  relative 
amount  of  cut  and  fill  required.  But  after  the  grade  has  been 
thus  chosen,  the  only  safe  method  in  side-hill  work  is  to 
actually  compute  the  amounts  of  excavation  and  embankment 
from  cross-sections,  mark  the  amount  for  each  cut  and  fill  on 
the  profile,  and  compare  the  results.  Any  changes  required  in 
the  grade  or  alignment  may  then  be  discovered  arid  effected 
before  the  work  of  construction  has  begun. 


CHAPTER    V. 

SIMPLE    CURVES. 
A.  Elementary  Relations. 

82.  The  centre  line  of  a  located  road  is  composed  alternately 
of  straight  lines  and  curves. 

The  straight  lines  are  called  tangents  because  they  are  laid 
exactly  tangent  to  the  curves.  A  tangent  may  be  indefinitely 
long,  but  should  never,  as  a  rule,  be  shorter  than  200  feet 
between  two  curves  which  deflect  in  opposite  directions,  nor 
shorter  than  500  feet  between  curves  which  deflect  in  the  same 
direction.  A  curve  should  not  be  less  than  200  feet  long. 
When  a  tangent  is  said  lo  be  straight,  the  meaning  simply  is 
that  it  has  no  deflections  to  the  right  or  left;  for  since  it  fol- 
lows the  surface  of  the  ground,  it  evidently  has  as  many 
undulations  as  the  ground.  But  if  we  conceive  a  vertical 
plane  to  be  passed  through  the  line,  a  horizontal  trace  of  this 
plane  will  accurately  represent  the  line;  and  so,  if  we  con- 
ceive a  vertical  cylinder  to  be  passed  through  a  curve  on  the 
surface  of  the  ground,  a  horizontal  trace  of  that  cylinder  will 
accurately  represent  the  curve,  since  all  distances  and  angles 
are  measured  horizontally,  whatever  be  the  irregularities  of 
the  surface.  In  all  problems,  therefore,  relating  to  this  sub- 
ject, we  may  consider  the  ground  to  be  an  absolutely  level 
plain. 

83.  A  Simple  curve  is  a  circular  arc  joining  two  tan- 
gents.   It  is  always  considered  as  limited  by  the  two  tangent 


SIMPLE   CURVES.  43 

points,  and  any  part  of  it  beyond  these  points  is  called  the 
curve  produced.  The  first  tangent  point,  or  the  point  where 
the  curve  begins,  is  called  the  Point  of  Curve,  and  is  indicated 
by  the  initials  P.  G.  The  point  where  the  curve  ends,  and  the 
next  tangent  begins,  is  called  the  Point  of  Tangent,  and  is  indi- 
cated by  the  initials  P.  T.  When  accessible,  these  points  are 
always  occupied  by  the  transit  in  the  course  of  the  survey, 
and  the  plug  driven  to  fix  the  point  is  guarded,  not  only  by 
the  usual  stake  bearing  the  number  of  the  station,  but  also  by 
another  bearing  the  proper  initials,  the  "  degree"  of  the  curve, 
and  an  "  R"  or  "L"  to  indicate  whether  the  deflection  is  to 
the  Eight  or  Left. 

84.  A  simple  curve  is  designated  either  by  the  radius,  JR, 
or  the  degree  of  curve,  D. 

The  Degree  of  Curve,  D,  is  an  angle  at  the  centre,  sub- 
tended by  a  chord  of  100  feet.  It  is  expressed  by  the  number 
of  degrees  and  minutes  in  that  angle,  or  in  the  arc  of  the 
x  curve  limited  by  the  chord  of  ICO 
feet.  Therefore  D  equals  the  num- 
ber of  degrees  of  arc  per  station. 

The  radius  R  and  degree  of 
curve  D  can  be  expressed  in  terms 
of  each  other. 

Let  ab,  Fig.  3,  be  a  chord  of 
Fl°-  3>  100    feet    subtending  an  arc   de- 

scribed with  a  radius  ao  —  R  from  the  centre  o.  Then,  by 
definition  the  angle  boa  —  D.  Bisect  the  angle  boa  by  a  line 
orj,  and  this  line  will  also  bisect  the  chord  ab  and  be  perpen- 
dicular to  it  ;  and  in  the  right-angled  triangle  bgo  we  have 

bg  =  ob  X  sin  bog 


Hence,  to  find  Radius  in  terms  of  Degree  of  Curve:. 

R=™  (16) 

sin  \D 

and  to  find  Degree  of  Curve  in  terms  of  Radius: 

sin  iD  =  A°  <17) 

JK 


44  FIELD   ENGIHEERLim 

It  is  the  practice  of  English  engineers  to  assume  the  radius 
at  some  round  number  of  feet  and  calculate  the  degree  of  curve, 
which  is  therefore  fractional.  In  America,  on  the  contrary, 
the  degree  of  curve  is  assumed  at  some  integral  number  of 
degrees  or  minutes,  and  the  radius  deduced  from  this. 

Example. — What  is  the  radius  of  a  3°  20'  curve? 

50  log        1.698970 

iD  =  l°40'        log  sin  8. 463665 

Ans.  ^  =  1719.12    log        3.235305 

Thus  the  second  and  third  columns  of  Table  IV.  have  been 
calculated. 

Example. — "What  is  the  degree  of  curve  when  the  radius  is 
600  feet? 

50  log        1.698970 

.8  =  600  log        2.7781^1 

\D  —  4°  46'  48". 73  log  sin  8.920819 
Ans.          D  =  9°  33' 37". 46 

Measurement  of  Curves. 

85.  A  railroad  curve  is  always  assumed  to  be  measured  with 
a  100-foot  chain,  and  as  the  chain  is  stretched  straight  between 
stations  it  cannot  coincide  with  the  arc  of  the  curve,  but 
forms  a  chord  to  the  arc,  as  in  Fig.  3.     Consequently  the 
curve  as  measured  from  one  tangent  point  to  the  other  is  an 
inscribed  polygon  of  equal  sides,  each  side  being  100  feet. 
The  sum  of  these  sides  (with  any  fraction  of  a  side  at  either 
end  of  the  curve)  is  called  the  Length  of  curve,  L.    This  length 
L  is  evidently  a  little  less  than  the  length  of  the  actual  arc 
between  the  same  points,  but  the  latter  we  very  seldom  have 
occasion  to  consider. 

86.  If  the  chain  lengths  were  taken  on  the  arc  instead  of  as 
chords  of  the  curve,  the  degree  of  curve  would  be  inversely 
proportional  to  the  radius,  and  since  the  arc  whose  length  is 
equal  to  radius  contains  57.3  degrees  nearly,  we  should  have 

D  :  57°. 3  ::  100  :  It. 
or 

573° 


SIMPLE   CURVES.  45 

a  convenient  formula,  but  only  approximately  true  when  D  is 
small,  and  seriously  at  fault  when  D  is  large;  the  error  in- 
volved being  proportional  to  the  difference  in  length  of  a 
100-foot  chord,  and  the  arc  which  it  subtends. 

87.  The  Central  Angle  of  a  simple  curve  is  the  angle 
at  the  centre  included  between  the  radii  which  pass  through  the 
tangent  points  (P.  C.)  and  (P.T.).  It  is  therefore  equal  to  the 
number  of  degrees  contained  in  the  entire  arc  of  the  curve 
between  those  points.  The  central  angle  will  be  designated 
by  the  Greek  letter  A  (delta). 

From  the  definitions  of  the  length  and  degree  of  curve  we 
have  the  proportion, 

D  :  A  ::  100  :  L. 

Hence,  to  find  the  Length  of  curve  in  terms  of  the  central 
angle: 

X  =  100  I  (18) 

Example.  —  What  is  the  length  of  a  4°  curve  when  the  cen- 
tral angle  is  29°  ? 

D  =  4°  and  A  =  29°  j  4)2900 

Ans.      L  =  7  stations  -f-  25  feet      I       725  feet. 

To  find  the  Central  angle  in  terms  of  the  length  and  degree 
of  curve: 

*  =  m  ™ 

Example.  —  What  is  the  central  angle  of  a  5°  curve  730  feet 
long? 


=  730,  =  36°.  5 

100 


Ans.  A  =  36°  30' 


To  find  the  Degree  of  curve  in  terms  of  the  length  and 
ventral  angle: 


Example.  —  What  is  the  degree  of  a  curve  8  stations  long, 
and  having  a  central  angle  of  26°  40'  ? 

L  =  800,         A  =  26°.666,         100  -^f6  =  3°.333 

oUU 

Ans.  D  =  3°  20' 


46 


FIELD   ENGINEERING. 


FIG.  4. 


88.  If  two  tangents,  joined  by  a  simple  curve,  are  produced 
(one  forward  and  the  other  backward)  until  they  intersect,  the 

point  of  intersection,  V  (Fig.  4), 
is  called  the  vertex,  and  the 
exterior  or  deflection  angle 
which  they  make  with  each 
other  is  equal  to  the  central 
angle,  A 

The  Tangent-distance, 
T,  is  the  distance  from  the 
vertex  to  either  tangent  point; 
thus  in  Fig.  4,  T=AV=VB. 
The  Long  Chord,  C,  is 
the  line  AB  joining  the  two 
tangent  points. 

The  Middle-ordinate, 
M,  is  the  line  QII,  joining  the 

middle  point  of  the  long  chord  with  the  middle  point  of  the 
curve. 

The  External  distance,  E,  is  the  line  HV,  joining  the 
middle  point  of  the  curve  with  the  vertex. 

We  observe  that  both  the  middle-ordinate,  M,  and  the 
external  distance,  E,  are  on  the  radial  line  joining  the  centre, 
0,  with  the  vertex,  V,  and  that  this  line  is  perpendicular  to 
the  long  chord,  C;  also,  that  it  bisects  the  central  angle 
AOB=  A,  and  its  supplement  A  VB.  (Tab.  1. 14.)  We  also 
observe  that  the  angle  VAB  =  VBA  =-}A  (Tab.  I.  20);  and 
if  in  the  figure  we  draw  the  two  chords  AH  and  BH,  the 
angle  BAH  equals  one  half  the  angle  BOH,  or  BAH—  ABU— 
i-A  (Tab.  I.  18);  also  the  angle  VAH=  VBH=±A. 

89.  If  we  have  laid  out  two  tangents  on  the  ground,  inter- 
secting at  V,  and  have  measured  the  angle,  A  ,  between  them, 
we  may  then  assume  any  other  one  of  the  elements  of  a 
simple  curve  before  mentioned,  and  calculate  the  rest.     If 
we  assume  D,  for  instance,  we  then  find  It  by  eq.  (16)  or  by 
Table  IV. 

Then,  having  A  and  E,  we  may  proceed  to  calculate  the 
other  elements  as  they  are  needed, 

90.  To  find  the  Tangent-distance    in  terms  of  the 
Radius  and  Central  Angle  : 


SIMPLE   CUKVES.  47 

In  the  right-angled  triangle  VGA,  Fig.  4,  we  have 

VA=OAX  tan  VOA 
.•:  T    =  .Rtan  -JA  (21) 

Otherwise,  approximately:  In  Table  VI.,  opposite  the  central 
angle,  take  the  value  of  T  for  a  1°  curve  and  divide  it  by  the 
degree  of  curve  D.  If  desirable,  add  the  correction  taken 
from  Table  V.,  corresponding  to  D. 

Example. — What  is  the  tangent  distance  of  a  4°  curve  with 
a  central  angle  of  30°  ? 

D  =  4°  R  (Table  IV.)     log  3 . 156151 

A  =  30°,         i  A  =  15°      log  tan  9.428052 

Ana.   T  =  383. 89  feet  log         2.584203 

Otherwise : 

By  Table  VI.  4)1535.3 

Approximate  ans.  383.82 

Correction  from  Table  V.  .08 


Ans.    T—  383.90  feet. 

91.  To  find  the  Long  Chord  C,  in  terms  of  Radius  and 

Central  Angle  : 
In  the  right-angled  triangle  BOG>  Fig.  4,  we  have 

BG  =  BOX  sin  BOG  ' 


.  C  =  2EsmiA  (22) 

But  in  case  A  can  be  divided  by  D  without  a  remainder, 
that  is,  if  the  curve  contains  an  exact  number  of  stations  (not 
exceeding  12),  we  may  take  the  long  chord  at  once  from 
Table  VII. 

Example.  —  What  is  the  long  chord  of  a  3°  20'  curve  with  a 
central  angle  of  36°  40'  ? 

2  log        0.301030 

D  -  3°  20',  R  (Tab.  IV.)    log         3.235305 
A  =  36°  40',  i  A  =  18°  20'  log  sin  9.497682 

Ans.  C  =  1081.48  feet  log        3.034017 


48  FIELD 

Otherwise  : 

O/»<>  • 

£  =  ?!-  =  It  stations 
2}       3J- 

And  by  Table  VII.  C=  1081.48. 

92.  To  find  the  Middle-ordinate  M,  in  terms  of  Radius 
and  Central  Angle: 

It  is  evident  from  the  figure  that  if  the  radius  OH  wore 
unity,  the  line  OH  would  be  the  nat.  versed  sine  of  the  arc 
BH.  But  the  arc  EH  measures  the  angle  JBOH=i&,  and 
OH=  R; 

A  (23) 


But  in  case  A  can  be  divided  by  D  without  a  remainder, 
that  is,  if  the  curve  contains  an  exact  number  of  stations  (not 
exceeding  12),  we  may  take  the  middle-ordinate  at  once  from 
Table  VIII. 

Example.  —  What  is  the  middle-ordinate  of-  a  4°  30'  curve 
with  a  central  angle  of  40°  30'  ? 

D  =   4°  30',        J2(Tab.  IV.)  log  3.105022 

A  =  40°  30',        |A  =  20°  15'  log  vers  8.791049 

Ans.  M  =  78.  717  1  .  896071 

Otherwise  : 

A      40.5      .    4 
'  5  =  -475  =  9  stations 

and  by  Tab.  VIII.  M  =  78.717 

93.  To  find  the  External  Distance  E  in  terms  of 
Radius  and  Central  Angle. 

It  is  evident  from  the  figure  that  if  the  radius  OA  were 
unity,  the  portion  HV  of  the  secant  line  0V  would  be  the 
external  secant  of  the  arc  AH.  But  the  arc  AH  measures  the 
angle  AOH—  £A,  and  OA  =  R; 

.'.  E  =  Rex  sec^A  (24) 

Otherwise,  approximately: 

In  Table  VI.,  opposite  the  central  angle,  take  the  value  of 
E  for  a  1°  curve,  and  divide  it  by  the  degree  of  curve  D. 
If  desirable,  add  the  proper  correction  corresponding  to  D, 
taken  from  Table  V. 


SIMPLE   CURVES.  49* 

Example. — What  is  the  external  distance  JEJ  of  a  7°  30'  curve 
when  the  central  angle  is  60°  ? 

D  =  7°  30',        R  (Tab.  IV.)  log  2 . 883371 

A  =  60°,  i  A  =  30°        log  ex  sec  9 . 189492 

Am.  E  =  118.27  feet  log  2.072863 

Otherwise: 

By  Tab.  VI.  7.5)886.38 

Approximate  ans.  118.184 

Correction  for  D  =  7°  30'  (Tab.  V.)  .084 

Ans.  E  =  118.268 

94.  But,  instead  of  assuming  D  or  R,  we  may  prefer,  or  may 
find  it  necessary  to  assume,  some  other  element  of  the  curve, 
the  central  angle  being  given. 

If  we  assume  the  tangent  distance,  then: 

95.  To  find  the  Radius  and  Degree  of  Curve  in  terms 
of  the  Tangent-distance  and  Central  Angle. 

From  eq.  (21),  and  by  Table  II.  40,  we  have 

E=Tcot^A  (25) 

Otherwise,  approximately: 

Divide  the  tangent  of  a  1°  curve  found  opposite  the  value  of 
A  in  Table  VI.,  by  the  assumed  tangent  distance;  the 
quotient  will  be  the  degree  of  curve  in  degrees  and  decimals. 

Example. — The  exterior  angle  at  the  vertex  is  54°,  and  the 
tangent  distance  must  be  about  700  feet.  What  shall  be  the 
degree  of  curve? 

A  =  54°,        $  A  =  27°      log  cot  0 . 292834 
T  -700  2.845098 


log£=  3.137932 

Am.  By  Table  IV.  D  =  4°  10'  + 

Otherwise: 

By  Table  VI.     700)2919.4 
Ans.  D  =  4°  10'  15"  4.1706 

But  as  it  is  difficult  to  lay  out  a  curve  when  D  is  fractional, 
we  discard  the  fraction  and  assume  4°  10'  as  the  value  of  Z>. 


•50 


FIELD 


This  may  require  us  to  recalculate  the  value  of  T,  which  we 
do  by  eq.  (21)  and  find  T  =  700.8  feet  log  2.845596.     If  the 
other  elements  are  required,  they  may  be  calculated  by  eqs. 
(22),  (23),  (24),  or  directly  from  Tand  A,  as  follows: 
96.  To  find  the  External  distance  E,  in  terms  of  the 
Tangent-distance  and  Central  Angle. 
In  Fig.  5  we  have  given 
AOB=  A  and  AV  —  T,  to  find 
HV=  E.      In    the    diagram  draw 
the  chord  AH,  and  through  //  draw 
a  tangent  line  to  intersect  OA  pro- 
duced in  /,  and  join  VI. 

Then  HI  is  parallel  to  BA,  and 
since HI=  AV=  T,  and  AI=  HV 
Fio  g  =  E,    VI  is  parallel   to   HA,   and 

VIH  =  HAB  =  i A.     (Tab.  I.  18.) 
In  the  right-angled  triangle  VHIwc  have 


tan  VIH 


or 


E— 


(26) 


Example. — The  angle  at  the  vertex  being  54°  and  the  tan- 
gent-distance" 700.80  feet,  how  far  will  the  curve  pass  from 
the  vertex  ? 

T=  700.80  (from  last  example)  2.845596 
A  —  54°5  i  A  =13°  30'   log  tan  9 . 380354  . 


Am.    E  =  168.25  feet 


2.225950 


(For  the  formulae  by  which  to  find  the  long  chord  and  mid- 
dle-ordinate  in  terms  of  the  tangent-distance  and  central  angle, 
see  Table  III.  12  and  13.) 

97.  Again,  it  may  be  necessary  to  assume  the  external  dis- 
tance in  order  to  determine  the  proper  degree  of  curve. 

To  find  the  Radius  and  Degree  of  Curve  in  terms  of 
the  External  distance  and  Central  Angle: 
k'  By  eq.  (24) 


E 


ex  sec 


(27) 


SIMPLE    CURVES.  51 

Otherwise: 

In  Table  VI.  divide  the  external  distance  of  a  1°  curve, 
opposite  the  given  value  of  A,  by  the  assumed  external  dis- 
tance ;  the  quotient  is  the  degree  of  curve  required. 

Example.—  The  angle  at  the  vertex  being  24°  30',  the  curve  is 
desired  to  pass  at  about  65  feet  from  the  vertex.  What  is  the 
proper  degree  of  curve  ? 

E  =  Q5         log  1.812913 

A  =  24°  30',         iA  =  12°  15'  log  ex  sec  8.367345 

logJ2=  3.445568 

Am.  By  Table  IV.  D  =  2°  03'  + 
Otherwise: 

By  Table  VI.     65)133.50 
Ans.  D  =  2°03'  14"  2°. 0538 

We  may  therefore  assume  a  2°  curve,  unless  required  by 
the  circumstances  to  be  more  exact,  when  we  might  use  a 
2°  03'  curve.  Assuming  a  2°  curve,  we  have  by  eq.  (24) 

.0=68.75    log  1.824460 

Having  decided  on  the  degree  of  curve,  we  may  calculate 
the  remaining  elements  by  eqs.  (21),  (22),  (23),  which  is  always 
the  better  way,  but  we  may  calculate  them  directly  from  E 
and  A. 

98.  To  find  the  Tangent-distance  in  terms  of  the 
External  distance  and  Central  Angle: 

From  eq.  (26),  and  by  Table  II.  40, 

T=Eco\,±&  (28)   • 

Example. — The  angle  at  the  vertex  is  24°  30',  and  the  curve 
passes  66.75  feet  from  the  vertex.  How  far  are  the  tangent 
points  from  the  vertex  ? 

E  -  68 . 75  (from  last  cxampls)         log        1 . 824460 
A  =  24°  30',     i  A  =  6°  07'  30"          log  cot  0.969358 


Ans.  T  =  622.04» feet  2.793818 

99.  Remark.— Eqs.  (27)  and  (28)  are  particularly  useful  in 
denning  the  curve  of  a  railroad  track  where  all  original 


FIELD   ENGINEERING. 


points  are  lost.  Produce  the  centre  lines  of  the  tangents  of 
the  curve  to  an  intersection  V,  and  there  measure  the  angle  A  . 
Bisect  its  supplement  A  VB,  and  measure  the  distance  on  the 
bisecting  line  from  Fto  the  centre  line  of  the  track.  This 
will  give  VII —  E.  Then  R  and  T  may  be  calculated,  and  the 
distance  T  laid  off  from  Fon  the  tangents,  giving  the  tangent 
points  A  and  B. 

(For  the  formulae  by  which  to  find  the  long  chord  and  mid- 
dle-ordinate  in  terms  of  E  and  A,  see  Table  III.  16  and  17.) 

100.  Again,  having  only  the  central  angle  given,  we  may 
assume  the  long  chord,  or  the  middle-ordinate,  and  from  either 
of  these  and  the  central  angle  calculate  the  remaining  ele- 
ments.    Or,  finally,  the  central  angle  being  unknown,  we  may 
suppose  any  two  of  the  linear  elements  given,  and  from  these 
calculate  the  rest.     As  such   problems  have  little  practical 
value,  their  discussion  is  omitted.     The  requisite  formulae  for 
their  solution  are  given  in  Table  III.,  and  the  verification  of 
them  is  suggested  as  a  profitable  exercise  to  the  student. 

B,  Location  of  Curves  by  Deflection  Angles. 

101.  In  order  that  the  stakes  at  the  extremities  of  the 
100-foot  chords,  by  which  the  curve  is  measured,  shall  be  set 

exactly  on  the  arc  of  the  curve 
by  transit  observation,  it  is  neces- 
sary at  the  point  of  curve,  A,  to 
deflect  certain  definite  angles 
from  the  tangent  AV.  Let  us 
suppose  that  in  the  curve  AB, 
Fig.  6,  the  points  A,  a,  b,  c,  d, 
etc.,  indicate  the  proper  posi- 
tions of  the  stakes  100  feet  apart, 
and  that  OA  is  the  radius  of  the 
curve.  In  the  diagram  join  Oa, 
Ob,  etc.,  and  also  Aa,  ab,  be,  etc. 
Then,  by  definition,  the  angle  AOa  =  D,  and  by  Geom. 
(Tab.  I.  20  and  11)  the  angle  VAa  =  ±D.  Therefore  if 
we  set  the  transit  at  A,  and  deflect  from  AV  the  angle 
\D,  we  shall  get  thexdirection  of  the  cltord  Aa,  on  which  by 
measuring  100  feet  from  A  we  fix  the  stake,  a,  in  its  true 
position  on  the  curve.  So  again,  since  the  angle  a  Ob,  at 
the  centre,  =  D,  the  angle  aAb,  at  the  circumference,  =  %D. 


FIG.  6. 


SIMPLE    CURVES.  53 

If  therefore,  with  the  transit  at  A,  we  deflect  the  angle  \I> 
from  the  chord  Aa,  we  shall  get  the  direction  of  the  chord 
Ab;  and  when  the  stake  b  is  on  this  chord  it  will  also  be  on 
the  curve,  if  b  is  100  feet  distant  from  a.  Thus,  in  general, 
we  may  fix  the  position  of  any  stake  on  the  curve,  by  deflect- 
ing an  angle  4Z>  from  the  preceding  stake,  and  at  the  same 
time  measuring  a  chain's  length  from  it, — the  chain  giving 
the  distance,  while  the  instrument  at  A  gives  the  direction  of 
the  point. 

\D  is  called  the  Deflection-angle  of  the  curve ;  so  that  in 
any  curve,  the  deflection-angle  is  equal  to  one  half  the  degree  of 
curve. 

102.  Since  each  additional  station  on  the  curve  requires 
an  additional  deflection-angle,  the  proper  deflection  to  be  made 
at  the  tangent  point  from  the  tangent  to  any  stake   on.  the 
curve  is  equal  to  the  deflection-angle  of  the  curve  multiplied 
by  the  number  of  stations  in  the  curve  up  to  that  stake ;  or  it 
is  equal  to  one  half  the  angle  at  the  centre  subtended  by  the 
included  arc  of  the  curve. 

103.  It  may  happen  that  all  the  stations  of  a  curve  are  not 
visible  from  the  tangent  point,  A.     When  this  is  the  case  a 
new  transit-point  must  be  prepared  at  some  point  on  the 
curve,  by  driving  a  plug  and  centre  in  the  usual  manner,  and 
the  transit  moved  up  to  it.     Let  us  suppose  that  the  point  d, 
Fig.  6,  has  been  selected  for  a  transit-point,    and  that  the 
transit  has  been  set  up  over  it.     Before  the  curve  can  be  run 
any  farther,  it  is  necessary  to  find  the  direction  of  a  tangent  to 
the  curve  at  the  point  d.     For  this  purpose  we  deflect  from 
chord  dA  an  angle  Adz  equal  to  the  angle  VAd  previously 
deflected  to  fix  the  point  d.    (Tab.  1. 16.)   Or  we  may  adopt  the 
following 

Rule :  To  find  the  direction  of  the  tangent  to  a 
curve  at  the  extremity  of  a  given  chord,  deflect  from  the  chord  an 
angle  equal  to  one  half  the  angle  at  the  centre  subtended  by  the 
chord.  (Tab.  I.  20.) 

Having  thus  found  the  direction  of  the  auxiliary  tangent 
zdx,  we  proceed  to  deflect  from-tfoj,  (^D)  for  the  next  station  e, 
2  (|D)  for  station/,  3(|D)  for  station  g,  etc.,  as  before.  When 
the  end  of  the  curve  is  reached,  a  transit-point  is  set  at  the 
Point  of  Tangent,  after  which  it  only  remains  to  find  the 
direction  of  the  tangent,  by  the  above  rule.  Thus  if  g  is  to  be 


54  FIELD 


the  point  of  tangent,  we  obtain  the  direction  of  the  tangent  by 
deflecting  from  the  chord  gd  an  angle  equal  to  xdy,  or  to 
\  dOg.  If  this  tangent  VB  was  already  established,  the  line 
gx  thus  obtained  should  coincide  with  it;  and  if  it  does  so, 
the  correctness  of  our  work  is  proved. 

104.  The  centre  line  is  measured,  and  the  stations  num- 
bered   regularly    and    continuously    through    tangents    and 
curves  from  the  starting  point  fo  the  end  of  the  work.     It 
therefore  frequently  happens  that  a  curve  will  neither  begin 
nor  end  at  an  even  station,  but  at  some  intermediate  point,  or 
plus  distance. 

If  the  Point  of  Curve  occurs  a  certain  number  of  feet 
beyond  a  station,  the  first  chord  on  the  curve  is  composed  of 
the  remaining  number  of  feet  required  to  make  100. 

Any  chord  less  than  100  feet  is  called  a  subchord. 

If  a  curve  ends  with  a  subchord,  the  remainder  of  the  100 
feet  must  be  laid  oft'  on  the  tangent  from  the  Point  of  Tangent 
to  give  the  position  of  the  next  station,  so  that  the  stations 
may  everywhere  be  100  feet  apart. 

105.  The  deflection  to  be  made  for  a  subchord  is  equal  to  one 
half  the  arc  it  subtends. 

Let  c  —  length  of  any  subchord  in  feet. 

"  d  =  angle  at  centre  subtended  by  subchord. 
Then,  from  eq.  (22),  by  analogy 

c  =  2.72  sin  $d  (29) 

100 
But  by  eq.  (16)  «Z  =  ' 


(30) 


.-.  smirf=~-sinp>  (31) 

When  D  does  not  exceed  8°  or  10°,  we  may  assume  without 
serious  error  that  the  angles  are  to  each  other  as  their  sines, 
and  the  last  two  equations  become 


(approx.)  c  =  100  -  (32) 


SIMPLE    CURVES.  55 

and  irf=~(P>)  (33) 

In  curves  sharper  than  10°  per  station,  the  error  involved  in 
this  assumption  becomes  apparent  and  must  be  corrected. 

1OO.  If  curves  were  measured  on  the  actual  arc,  then 
eqs.  (32)  and  (33)  would  be  true  in  all  cases;  but  since  a  curve 
is  measured  by  100-ft.  chords,  it  is  evident  that  if  a  100-ft. 
chord  between  any  two  stations  were  replaced  by  two  or  more 
subchords,  these  taken  together  would  be  longer  than  100  feet, 
since  they  are  not  in  the  same  straight  line.  Let  us  conceive 
the  actual  arc  of  one  station  to  be  divided  into  100  equal 
parts;  since  the  arc  is  longer  than  the  chord,  each  part  will  be 
slightly  longer  than  one  foot.  Now  if  we  take  an  arc  contain- 
ing any  number  of  these  parts  (less  than  100),  the  nominal 
length  of  the  corresponding  subchord  in  feet  will  equal  the 
number  of  parts,  and  the  deflection  for  the  subchord  will  be 
proportional  to  the  number  of  parts  which  the  arc  contains. 
The  deflection  therefore  will  be  exactly  given  by  eq.  (33)  if  in 
that  equation  we  let  c  equal  the  number  of  parts  in  the  arc,  or 
the  nominal  length  of  the  subchord  in  feet.  Having  thus 
obtained  the  correct  value  of  (%tf),  we  may  introduce  it  into 
eq.  (29)  or  (30),  and  obtain  the  true  value  of  the  subchord, 
which  will  always  be  a  little  greater  than  its  nominal  value. 

Suppose,  for  instance,  that  the  arc  of  one  station  is  to  be 
divided  into  four  equal  portions;  then  each  subchord  will  be 
nominally  25  feet  long;  and  by  cq.  (33) 


which  is  the  correct  value  of  the  deflection,  whatever  be  the 
degree  of  curve.  Substituting  this  value  in  eq.  (29)  .or  (30)  we 
obtain  the  true  value  of  the  subchord,  c,  a  little  greater  than 
25;  the  excess  is  called  the  correction  of  the  nominal  length. 

1O7.  This  correction  for  any  given  subchord  bears  an 
almost  constant  ratio  to  the  excess  of  arc  per  station,  what- 
ever be  the  degree  of  curve.  These  ratios  are  shown  in  the 
following  table  for  a  series  of  subchords,  and  Table  VII.  gives 
the  length  of  actual  arc  per  -station  for  various  degrees  of 
curve.  Subtracting  100  we  have  the  excess  of  arc  per  station, 
and  multiplying  this  excess  by  the  ratio  corresponding  to  the 


56 


FIELD   ENGINEERING. 


nominal  length  of  subchord  we  obtain  as  a  product  the  proper 
correction  for  the  subchord. 

TABLE  OP  THE  RATIOS  OF  CORRECTIONS  OF  SUBCHORDS  TO 
THE  EXCESS  OF  ARC  PER  STATION. 


Nominal 
Length  of 
Subchord. 

Ratio. 

Nominal 
Length  of 
Subchord. 

Ratio. 

Nominal 
Length  of 
Subchord. 

Ratio. 

0 
5 
-    10 
15 
20 
25 
30 

.COD 
.059 
.099 
.147 
.192 
.234 
.273 

35 
40 
45 
50 
55 
60 
65 

.507 
.33'J 
.358 
.374 
.383 
.383 
.374 

70 
75 
80 
85 
90 
95 
100 

.356 
.327 

.287 
.235 
.169 
.OJ2 
.003 

We  observe  that  the  largest  correction  is  required  by  a  sub- 
chord  between  55  and  60  feet  in  length. 

Example. — It  is  proposed  to  run  a  14°  curve  with  a  50-ft. 
chain.  What  correction  must  be  added  to  the  chain? 


By  eq.  (30) 

c  =  100 
Ans.  Correction  =  .093 

Or,  by  Table  VII., 


and  by  above  table, 

Ans.  Correction  =  product 


=  ~  X  7°  =  3°.5  =  3°  30' 


=  50.093 

sm  7 


length  of  arc  =  100.249 
excess  of  arc  =  .  249 
ratio  for  50  feet  =  .  374 


=        .093 


Example.  —  The  P.  C.  of  an  18°  curve  is  fixed  at  -}-  55  feet 
beyond  a  station.  What  are  the  nominal  and  true  values  of 
the  first  subchord,  and  what  the  proper  deflection? 

Nominal  value  =  100  —  55  =  45  feet 


Deflection  =  %d  =     -  X  9°  =  4°.  05  =  4°  03' 

1UU 


and  by  eq  (30) 


True  value  =  c  =  100 


sm 


.    ft 
sm9 


-  =  45.148 


SIMPLE   CURVES.  57 

Or,  by  Table  VII.,  excess  of  arc  —      .412 

by  above  table,  ratio  for  45  feet  =      .  358 

Correction  =  product  =      .  147 
Ans.  True  value  of  subchord  =  45 . 147 

Example. — The  last  deflection  at  the  end  of  a  40°  curve  is 
found  to  be  6°  30'.  What  are  the  nominal  and  true  values  of 
the  last  subchord? 

Here  $d  =  6°  30',  and  by  eq.  (32) 

/»     p* 

Nominal  value,  c  =  100  -^-  =  32.5  feet 
By  eq.  (30) 

True  value,  c  =  100  ~-~    ~  =  33.098  feet 
sin  20 

Or  by  Table  VII.,  excess  of  arc  40°  =    2.060 

by  above  table,  ratio  for  32 . 5  feet  =      .290 

Correction  —  product  =      .597 
Nominal  value  of  subchord  =  32 . 5 


True  value  =33.097 

1O8.  For  convenience  in  making  deflections,  the  zeros  of 
the  instrument  should  always  be  together  when  the  line  of 
collimation  coincides  with  a  tangent  to  the  curve.  Thus,  in 
beginning  a  curve,  the  transit  being  set  at  the  P.  C.  zeros 
together,  and  line  of  collimation  on  the  tangent,  the  read- 
ing of  the  limb  for  any  station  on  the  curve  has  simply  to  be 
made  equal  to  the  proper  deflection  from  the  tangent  for  that 
station.  After  the  transit  is  moved  forward  from  the  P.  C. 
and  set  at  another  point  of  the  curve,  the  vernier  is  set  to  a 
reading  equal  to  the  reading  used  to  establish  that  point,  but 
on  the  opposite  side  of  the  zero  of  the  limb,  and  the  line  of 
collimation  is  set  on  the  P.  C.  just  left.  Then  by  simply  turn- 
ing the  zeros  together  again,  the  line  of  collimation  will  be 
made  to  coincide  with  a  tangent  to  the  curve  through  the  new 
point,  and  the  deflections  for  the  succeeding  stations  can  be 
read  off  directly,  as  before.  Thus  any  number  of  transit 
points  may  be  used  in  locating  a  curve  by  finding  the  direc- 
tion of  the  tangent  through  each  by  a  deflection  from  the  pre- 
ceding point,  until  finally  the  P.  T.  is  reached,  where  another 
deflection  gives  the  direction  of  the  located  tangent. 


58 


FIELD   ENGINEERING. 


1O9.  The  assistant  engineer  keeps  neat  and  systematic 
field-notes  of  all  his  operations  with  the  transit  in  running 
curves.  The  numbers  of  the  stations  are  written  in  regular 
order  up  the  first  column  of  the  left-hand  page  of  the  field- 
book,  using  every  line,  or  every  other  line,  as  may  be  pre- 
ferred. The  second  column  contains  the  initials  of  each 
transit  point  on  the  same  line  as  the  number  of  its  station,  or 
between  lines,  if  the  point  occurs  between  two  stations.  In 
the  third  column,  and  opposite  the  initials  in  the  second,  is 
recorded  the  station  and  plus  distance,  if  any,  of  each  transit 
point.  The  fourth  column  contains,  opposite  the  "P. (7.,"  the 
degree  of  curve  used,  and  an  R  or  L,  showing  whether  the 
curve  deflects  to  the  right  or  left ;  the  fifth  column  contains 
the  readings  or  deflections  made  from  a  tangent  to  set  each 
station  or  point,  written  on  the  same  line  as  the  number  of 
that  station  or  point;  and  the  sixth  column  contains  the  cen- 
tral angle  of  the  whole  curve,  A,  written  opposite  the  "P. T" 
The  plus  distances  recorded  in 
the  third  column  are  always  the 
nominal  lengths  of  subchords,  but 
if  the  true  lengths  have  been  calcu- 
lated and  laid  off  on  the  ground, 
these  should  also  be  recorded  in 
parenthesis.  On  the  right-hand 
page  are  recorded  the  calculated 
bearings  of  the  tangents  and  their 
A  magnetic  bearings;  and  on  the 
centre  line  of  the  page,  opposite 
the  record  of  each  transit  point,  a 
dot  is  made  with  a  small  circle 
around  it,  to  show  the  relative  position  of  the  several  points 
on  the  ground.  Some  slight  topographical  sketches  may  be 
made,  indicating  the  more  prominent  objects,  but  the  full 
sketches  should  be  taken  by  the  topographer  in  a  separate  book. 
HO.  Since  the  deflections  start  from  zero  at  each  new 
transit  point,  the  sum  of  the  deflections  by  which  the  transit 
•points  are  located  will  be  equal  to  one  half  the  central 
angle  of  the  curve. 

111.  The  stations  on  a  curve  may  be  located  by  deflec- 
tions only,  Avithout  linear  measurements.  For  this  purpose 
two  transits  are  set  at  two  transit  points  on  the  curve,  as  A 


FIG.  7. 


SIMPLE   CURVES. 


59 


and  B,  Fig.  7,  and  the  proper  deflections  for  any  station  are 
made  with  both  instruments,  the  station  being  located  by  find- 
ing the  intersection  of  the  two  lines  of  collimation. 

This  method  requires  that  the  two  transit  points  shall  have 
been  previously  established,  that  their  distance  from  each 
other  shall  be  known,  that  they  shall  be  visible  from  each 
other,  and  that  they  shall  both  command  a  view  of  the  stations 
to  be  located.  It  is  not  therefore  generally  useful,  but  may 
be  resorted  to  to  set  stations  which  fall  where  chaining  cannot 
be  accurately  done,  as  in  water  or  swamps.  The  chord  join- 
ing the  two  transit  points  becomes,  in  fact,  a  base-line,  and  the 
deflections  form  a  series  of  triangulations. 

C.  Location  of  Curves  by  Offsets. 

112.  A  curve  may  be  located  by  linear  measurement  only, 
without  angular  deflections.  There  are  four  general  methods, 


By  offsets  from  the  chords  produced, 
By  middle-ordinates, 
By  offsets  from  the  tangents,  and 
By  ordinates  from  a  long  chord. 

To  locate  a  curve  by  offsets    from  the  chords 
produced. 

When  the  curve  begins  and  ends  at  a  station. 

113.  Let  A,  Fig.  8,  be  the  P.  C.  of  a  curve  taken  at  a  station, 
to  locate  the  other  stations,  a,  b,  c, 
etc.  The  chords  Aa,  ab,  be,  etc., 
each  equal  100  feet,  and  since  the 
angle  AOa  =  D,  the  angle  VAa  — 
\D.  (Tab.  I.  20.)  Taking  an  off- 
set ax  —  t,  perpendicular  to  the 
tangent,  we  have  in  the  right- 
angled  triangle  Axa. 

ax  =  Aa  X  sin  \T> 
or 

t    =  100  sin  ^D        (34) 
The  offset  t  is  called  the  tangent 
offset,  and  its  value  is  givenfor  all 
degrees  of  curve  in  Tab.  IV.  col.  4.  FIG.  8. 

If    the    curve    were    produced 
backward  from  A,  100  feet  to  station  z,  the  offset  zy  would 


60  FIELD 

equal  t;  and  if  the  chord  zA  were  produced  100  feet  from  A 
to  a',  the  offset  a'x  would  also  equal  t.  Therefore  the  distance 
aa'  =  2t,  and  the  angle  aAa'  =  D.  So  if  we  produce  the  chord 
Aa  100  feet  to  b',  the  distance  bb'  =  2t. 

To  lay  out  the  curve,  stretch  the  chain  from  A,  keeping  the 
forward  end  at  a  perpendicular  distance,  t,  from  the  line  of  the 
tangent  to  locate  station  a.  Then  find  the  point  b'  by  stretch- 
ing the  chain  from  a  in  line  with  a  and  A,  and  then  stretching 
the  chain  again  from  a,  fix  its  forward  end  at  a  distance  from 
b'  equal  to  2t.  This  gives  station  b.  In  the  same  way  find 
other  stations. 

When  the  last  station,  as  d,  of  the  curve  is  reached,  produce 
the  curve  one  station  farther  to  c" '.  Then  the  tangent  through 
d  is  parallel  to  the  chord  ce",  and  laying  off  t  from  c  and  e"  per- 
pendicular to  this  chord,  the  tangent  c"e  is  found.  If  the  work 
has  been  correctly  done  the  tangent  c"e  will  coincide  with  the 
given  tangent  VB. 

When  the  curve  begins  or  ends  with  a  subchord. 

114.  Let  A,  Fig.  9,  be  the  P.O.  and  Aa  the  first  sub- 
chord  =  c,  and  the  angle  VAa  =  ^d,  and  let  the  offset  ax  =  ^. 

Then 

ti  =  c  sin  ^d  (35) 

Producing  the  curve  backward  to  the  nearest  station  z,  we 
have  another  subchord  Az  =  (100  —  c),  and  the  angle  yAz  =  i 
(D  —  d)t  and  putting  the  offset  yz  =  t, 

t.  =  (100  -  c)  sin  i(D-d)  (36) 

Laying  off  the  two  subchords  on  the  ground,  and  making 
the  proper  offsets,  tt  and  tu,  at  the 
same  time,  we  fix  the  position  of 
the  two  stations  a  and  z  on  the 
curve  ;  after  which  we  may  pro- 
duce the  chord  za  100  feet  to  b', 
and  proceed  as  before  until  the 
curve  is  finished. 

If  the  curve  ends  with  a  sub- 
chord,  as  dB,  produce  the  curve 
to  the  first  station  beyond  B,  as 
e",  then  calculate  the  two  offsets 
for  the  two  subchords  Bd  and  Be", 
and  lay  them  off  from  d  and  e" 


SIMPLE   CUBVES.  61 

perpendicular  to  the  supposed  direction  of  the  tangent.  If 
the  line  d"e  so  obtained  coincides  with  the  given  tangent,  VB, 
the  work  is  correct. 

115.  We  may  find  the  values  of  tt  and  tu  otherwise  than 
by  the  formulae  above,  for  in  Fig.  8  we  have  shown  that  the 
angle  aAoJ  —  aOA,  and  since  these  triangles  are  isosceles, 
they  are  similar;  therefore 

Fig.  8,  OA  :  Aa ::  Aa  :  aa' 

or  R:  100::  100  :  2t 


and  similarly,  Fig.  9, 

t  -  •  — 

Hence 

t,  :  t  ::  c2  :  (100)2 

c2  t 


Thus  tt  may  be  found  by  multiplying  the  square  of  the  sub- 
chord  by  the  value  of  t  given  in  Tab.  IV.,  and  dividing  the 
product  by  10000.  As  c  is  always  less  than  100,  so  t,  is  always 
less  than  t. 

116.  In  eqs.  (35),  (38),  and  (39)  it  is  customary  to  use  the 
nominal  values  of  c,  and  this  can  produce  no  error  in  t  or  t, 
exceeding  -005,  when  the  degree  of  curve  does  not  exceed  ten 
degrees.    In  the  case  of  a  very  sharp  curve,  the  formulae  eqs. 
(40)  and  (41)  are  preferable. 

To  locate  a  curve  by  middle-ordinates. 

"When  the  curve  begins  and  ends  at  a  station. 

117.  In  Fig.  10,  let  A  be  the  P.  C.  at  a  station,  and  let  a  and 
z  be  the  next  stations  on  the  curve  either  way  from  A.     Then, 
since  zy  =  ax  =  t,  the  chord  za  is  parallel  to  the'  tangent  A  V, 
and  Ag  =  t.     Hence,  having  any  two  consecutive  stations  on 
the  curve,  as  z  and  A,  we  may  lay  off  the  tangent  offset  t 
from  A  to  g  on  the  radius,  and  find  the  next  station,  a,  100  feet 
from  A  on  the  line  zg  produced.     Then  laying  off  ah  =  t  on 
the  radius  aO,  a  point  on  the  line  Ah  produced  and  100  feet 
from  a  will  be  the  next  station  b. 


62 


FIELD 


On  reaching  the  end  of  the  curve,  the  tangent  is  found 
precisely  as  described  in  the  method  by  chords  produced,  §  113. 

In  Fig.  10,  we  observe  that  if  the  radius  OA  were  unity,  gA 
would  be  the  versed  sine  of  the  angle  aOA  =  D.  But  gA  =  t, 


.  t  =  R  vers  D 


(40) 


When  the  curve  begins  or  ends  with  a  subchord. 

118.  Let  A,  Fig.  11,  be  the  P.O.,  and  a  and  z  the  nearest 


FIG.  10. 


FIG.  It 


stations.     Then  Aa  =  c,  the  first  subchord,  and  aOA  =  d,  and 
by  analogy,  we  have  from  the  last  equation,  if  ax  =  t,  and 


t,  =  R  vers  d 

t   = 


(41) 


or  eq.  (39)  may  be  used  if  preferred. 

Having  found  the  two  stations,  a  and  z,  on  the  curve,  lay 
off  from  the  forward  station  a,  ah  =  t  on  the  radius,  and  so 
continue  the  curve  as  described  above. 

When  the  end  of  the  curve  is  reached,  produce  the  curve  to 
the  next  station  beyond,  and  find  the  tangent  by  offsets  as 
described  in  the  previous  method,  §  114. 

To  locate  a  curve  by  offsets  from  the  tangents. 

When  the  curve  begins  at  a  station. 

119.  Let  A,  Fig.  12,  be  the  P.O.  at  a  station.  Then  the 
next  station  a  is  located  by  the  tangeot  offset  t,  taken  from 


SIMPLE   CURVES. 


63 


Tab.  IV.,  or  calculated  by  eq.  (40).  To  calculate  the  distances 
and  offsets  for  the  following  stations,  b,  c,  etc. ,  in  the  diagram 
draw  lines  through  the  points  b,  c,  etc.,  parallel  to  the  tangent 
AV,  intersecting  the  radius  AO  in  g' ,  g",  etc.,  and  draw  the 
lines  bx' ,  ex",  etc.,  perpendicular  to  the  tangent. 
Then 

Ax'  =  g'b  =  Ob  sin  bOA 


and 
Also, 

or 


and 


Ax'  =  R  sin  2Z> 
Ax"  =  It  sin  3D 
etc.  etc. 


bOA 


f  = 

t"  =R  vers  3Z) 

etc.          etc. 


(43) 


(43) 


But  these  calculations  may  be  avoided,  for  as  twice  ag  equals 
the  chord  of  two  stations,  so  twice  bg'  equals  the  chord  of  four 
stations,  and  twice  eg"  the  chord 
of  six  stations,  etc.    So  also  as  Ag 
is  the  middle-ordinate  of  two  sta- 
tion, Ag'  is  the  middle-ordinate  of 
four,  and  Ag"  the  middle-ordinate 
of   six  stations,  etc.     Hence  the 
rule : 

The  distance  on  the  tangent  from 
the  tangent  point  to  the  perpendicu- 
lar offset  for  the  extremity  of  any 
arc  is  equal  to  one  half  the  long 
chord  for  twice  that  arc;  and  the 
offset  from  the  tangent  to  the  ex- 
tremity of  any  arc  is  equal  to  the 
middle-ordinate  of  twice  that  arc. 

The  long  chords  and  middle-ordinates  may  be  taken  from 
Tables  VII.  and  VIII.  for  2,  4,  6,  8,  etc.,  stations,  when  the 
P. (7.  is  at  a  station,  or  for  1,  3,  5,  7,  etc.,  stations,  when  the 
P.  C.  is  at  +  50,  or  half  a  station. 

If  the  offsets  from  the  first  tangent  AV  prove  inconveniently 
long,  the  second  half  of  the  curve  may  be  located  from  the 
other  tangent  BY,  beginning  at  the  point  of  tangent  B,  and 
closing  on  a  station  located  from  the  first  tangent. 


FIG.  12. 


64 


FIELD 


When  the  curve  begins  with  a  subchord. 

12O.  If  d=the  angle  at  centre,  subtended  by  the  first 
subchord,  we  have  for  the  distances  on  the  tangent  (Fig.  13) 


Ax  =  .R  sin  d 
Ax'  =  R  sin  (d  +  Z>) 
Ax"  =  E  sin  (d  +  2i>) 
etc.  etc. 

and  for  the  offsets  (Fig.  11) 

t,  =  R  vers  d 
t  =R  vers  (d  +  D) 
t"  =  R  vers  (d  -f-  2D) 
etc.  etc. 


(44) 


(45) 


If  the  first  subchord  equals  50  feet  (nominal),  then  d  =^D, 
and   the  Tables  VII.  and  VIII.  may  be  used  as  explained 


FIG.  13. 


FIG.  14. 


above.  These  tables  may  be  used  in  any  case,  by  adopting  a 
temporary  tangent  through  any  station,  and  laying  off  the  dis- 
tances on  this,  and  making  the  offsets  from  it. 

When  a  curve  is  located  by  offsets  the  chain  should  be  car- 
ried around  the  curve,  if  possible,  to  prove  that  the  stations 
are  100  feet  apart. 

To  locate  a  curve  by  ordinates  from  a  long 
chord. 

WJien  the  curve  begins  and  ends  at  a  station. 

121.  In  Fig.  14  draw  the  long  chord  AB,  joining  the  tan- 
gent points,  and  from  this  draw  ordinates  to  all  the  stations  on 


SIMPLE   CURVES.  65 

the  curve.    We  then  require  to  know  the  several  distances  on 
the  long  chord  Aa',  a'b',  b'c',  etc.,  and  the  length  of  ordinate 
at  each  point. 
Let  G  =  the  long  chord  AB,  then  eq.  (22) 

C=2E  sin  |A 

If  a  is  the  second  station  and  *  next  to  the  last  on  the  curve, 
join  ai,  and  let  the  chord  ai  =  C'.  Then  since  the  arc  Aa  — 
ik  =  D,  the  angle  at  the  centre  subtended  by  €'  is  (  A  —  2D). 

.•.   (7' =  212  sin  i  (A -21?) 

Again,  if  we  join  b  and  h  the  next  stations  and  let  bh=  C' 
G"  =2R  sin  £  (A  -  4D) 

and  so  on  for  other  chords. 
Since  Aa'  =  ki,  C=  C'  +  2Aa' 

,  ,      C-G1 
•''Aa=-2- 
Similarly, 

,•       c'-cr 

a  V  — « 


Thus  we  continue  to  find  the  distances  up  to  the  middle  of 
the  curve,  after  which  they  repeat  themselves  in  inverse 
order. 

122.  When  the  long  chord  G,  subtends  an  even  number  of 
stations  (as  10  in  Fig.  14),  the  middle  ordinate  of  the  chord  is 
the  ordinate  of  the  middle  station,  as  e.  Since  the  chords  AB 
and  ai  are  parallel,  the  ordinate  a' a  or  Hi  is  evidently  equal  to 
the  difference  of  the  middle  ordinates  of  these  chords. 

Let  M,  M',  M",  etc.,  be  the  middle-ordinates  of  the  chords 
C,  G',  G",  etc.  Then  eq.  (23) 

M  =^vers|A 
M'  =  12vers|(A  -  2J9) 
M"  =  levers  £  (A  -  4D) 
etc. ,  etc. 

And  a' a  =  i'l  —  M  —  M ' 

b'b  =7i'h  =  M-M" 
etc.       etc.        etc. 

The  values  of  the  chords  and  middle-ordinates  may  be  taken 
at  once  from  Tables  VII.  and  VIII. 


66 


FIELD 


Example. — It  is  required  to  locate  a  4  degree  curve  of  ten 
stations  by  offsets  from  the  long  chord. 
By  Table  VII. : 

^Diff. 


97. 030  =  «'&'  =  i'h' 
98.481  =b'c'  =h'g' 
99.452  =  c'<f  =  g'f 
99.939  =  dV  =  fe' 


Diff. 

10  sta. 

C  =980.014 

190.211 

8  " 

(71  =789.803 

194.059 

6  " 
4  " 

(711  =  595.744 

(7m=398.782 

196.962 
198.904 

2  " 

(7^  =  199.878 

199.878 

0  " 

&  =000.000 

From  Table  VIII. 


Diff. 


10  sta. 

M     =86.402 

8     ' 

M'1     =55.500 

30.902 

=  a'  a  =  i'i 

6     ' 

Jf"    =31.308 

55.094 

=  Vb  =  h'h 

4     ' 

Jf"1  =13.943 

72.459 

=  c'c  =  g'g 

2     ' 

Jfiv  =    3.490 

82.912 

=  d'd  =  ff 

0     ' 

Jtfv   =    0.000 

86.402 

=  e'e 

123.  When  the  long  chord  C  subtends  an  odd  number  of 
stations,  the  middle  ordinate  will  fall  half-way  between  two 
stations,  and  need  not  be  laid  off. 

If  the  ordinates  near  the  middle  of  the  curve  prove  incon- 
veniently long,  we  may  subtract  M  —  M',  M'  —  M",  etc.,  and  so 
obtain  in  Fig.  14  a' a,  b"b,  c"c,  etc.  We  then  lay  off  Aa',  a' a, 
ab",  b'b,  be",  etc.,  turning  a  right  angle  at  every  point.  The 
chain  should  be  carried  along  the  curve  at  the  same  time  to 
make  the  stations  100  feet  apart. 

Example. — It  is  required  to  locate  a  10-degree  curve  of  nine 
stations  by  offsets  from  the  long  chord. 

By  Table  VII. : 


Diff. 


^Diff. 


9  sta.  811.314 

7  "  658.105 

153.209 

76.  604  =  Aa' 

5  "  484.900 

173.205 

86.  603  =  <*'&' 

3  "  296.962 

187.938 

93.969  etc. 

1  "  100.000 

196.962 

98.481 

0  "   0.000 

100.000 

50.000 

SIMPLE   CURVES. 


67 


By  Table  VIII. : 
9  sta.  168.029 


103.750 

53.750 

19.548 

-2.183 

0.000 


Diff. 
64.279 
50.000 
34.202 
17.365 
2.183 


=  a'a 

=  b"b 

=  c"c 

etc. 


124.  The  tables  can  be  used  equally  well  when  the  curve 
both  begins  and  ends  with  a  half  station  ;  also  to  locate 
half-station  points  throughout  the  curve,  but  in  the  latter  case 
the  numbers  are  taken  from  consecutive  columns  of  the  tables 
instead  of    from   alternate    col- 

umns, as  in  the  above  examples. 

When  tJie  curve  begins  or  ends 
with  any  subchord. 

125.  Let  A,  Fig.  15,  be  the 
P.  C.  and  Aa  =  c  the  first  sub- 
chord,  and  d  the  angle  it  sub- 
tends at  the  centre.     In  the  dia- 
gram draw  the  long  chord  AB, 
and  the  ordinates  to  each   sta- 
tion, and   through  each  station   O 
draw  a  line  parallel  to  AB,  and 
let  AOB=  A. 

Since  the  angle  VAB  =  \  A  and 

VAa  =  \d,  the  angle  a  AB  —  \  (  A  —  d).  The  deflection  angle 
from  the  subchord  Aa  produced  to  the  chord  ab  is  $  (d  -f-  D), 
the  deflection  angle  between  any  two  consecutive  chords  of 
100  feet  is  |  (D  -f  D  )  =  D.  Therefore  the  angle 


FIG.  15. 


bob"  -  $  (A  -  d)  - 


=  i  (A  -  2d  - 


«  =  *  (A  -  2rf-D)  -  i  (2.D)  =  i(  A  -  2d  - 
'  =  ±  (A  -  2d-W)  -  i  (2D)  =!(A  -  2<Z- 
etc.  etc.  etc. 


FIELD   E^GINEEKING. 


Solving  the  several  right-angled  triangles  we  have,  Fig.  15. 


And  also 


Aa'  =  c.     cos  |  ( A  —  d) 
ab"  =  100  cos  -H  A  —  2d  —   D) 
be"  =  100  cos  i  (A  -  2d -  3jL») 
dd"  =  100  cos  i  (  A  —  2d  —  5D) 
etc.,  etc., 

a' a  =  c.     sin  $  (  A  —  d) 
b"b  =  100  sin i(A  -  2d  -    D} 
c"c  —  100  sin |  (A  —  2d  —  3Z>) 
d"c  =  100  sin  i  (A  -  2d  -  5Z>) 
etc.,  "etc., 


(46) 


(47) 


When  the  middle  point  of  the  curve  is  passed  the  minus 
quantities  in  the  parentheses  become  greater  than  A,  making 
the  parentheses  negative,  and,  therefore,  the  sines  negative, 
and  indicating  that  such  values  as  are  determined  by  them 
must  be  laid  off  toward  the  long  chord  AB. 

By  a  proper  summation  of  the  quantities  determined  by  eqs. 
(46)  and  (47)  we  obtain  the  distances  Aa',  Ab',  Ac',  etc.,  and 
the  ordinates  a' a,  b'b,  c'c,  etc.,  and  the  curve  may  be  located 
accordingly.  It  is  well  to  make  all  the  necessary  calculations 
before  beginning  to  lay  down  the  lines  on  the  ground,  thus 
avoiding  confusion  and  mistakes. 

Example.— The  P.O.  of  a  3°  20'  curve  is  fixed  at  -|-  25  feet 
beyond  a  station,  and  the  central  angle  is  16°  24'  =  A.  It  is 
required  to  locate  the  curve  by  ordinates  from  the  long  chord. 

We  have  c  =  100  -  25  =  75  and  d  =  2°  30'  and  D  =  3°  20'. 
Hence,  eqs.  (46) 


Aa'=    75  cos        6°  57'    =74.449 

ab"  =  100  cos  4°  02'  =  99.752 
be"  =  100  cos  0°  42'  =  99.993 
d"d  =  100  cos  (-  2°  38')  =  99.894 
e"e  =  100  cos  (-  5°  58')  =  99.458 
e'B  =  17  cos  (-  7°  55')  =  16.838 

By  eqs.  (47) 

a'a=  75  sin  6°  57'  =  9.075 
&"&=100sin  4°  02'=  7.034 
c"c  =  100  sin  0°  42'  =  1 .222 
cd"  =  100  sin  (-  2°  38')  =  -  4.594 
de"  =  100  sin  (-  5°  58')  =  -  10.395 
ee'  =  17  sin(-  7°  55')  =  -  2.341 


74.449  =  Aa' 
174.201  =  Ab' 
274.194  =  ^' 
374.088  =  Ad' 
473.546  =  J.«' 
490.384  =  AB 


9.075  =  a' a 
16.109  =  b'b 
17.331  =  c'c 
12.737  =  d'd 

2.342  =  e'e 

0.000    . 


SIMPLE   CURVES. 


69 


The  same  formula?  can  be  used  when  the  curve  begins  at  a 
station  by  making  c  =  100  and  d  —  D. 

126.  The  methods  of  locating  curves  by  linear  measure- 
ments do  not  require  the  use  of  a  transit,  although  one  may 
be  used  to  advantage  for  giving  true  lines,  turning  right 
angles,  etc.  When  a  transit  is  not  used  the  alignments  should 
be  made  across  plumb-lines  suspended  over  the  exact  points 
previously  marked  on  top  of  the  stakes.  A  right  ang'le 
may  easily  be  obtained,  without  an  instrument,  by  laying  off 
on  the  ground  the  three  sides  of  either  of  the  right-angled 
triangles  represented  in  the  following  table  (or  any  multiples 
of  them),  always  making  the  base  coincide  with  the  given  line. 

TABLE  OF  RIGHT-ANGLED  TRIANGLES. 

Hypothenuse.  Perpendicular. 

5  3 

13  5 


4 
12 
24 
40 
60 
84 
96 


25 

41 

61 

85 

100 


7 

9 

11 

13 

28 


D.  Obstacles  to  the  Location  of  Curves. 

127.  To  locate  a  curve  joining  two  tangents  when  the  in- 
tersection V  is  inaccessible.    Fig.  16. 

From  any  transit.point  p  on  one  tangent  run  a  line  pq  to 
intersect  the  other  tangent;  measure 
pq  and  the  angles  it  makes  with  the 
tangents.  Then  the  sum  of  the  de- 
flections at  p  and  q  equals  the  central 
angle  A.  Solve  the  triangle  pqV 
and  find  Vp.  Having  decided  on 
the  radius  R  of  the  curve,  calculate 
the  tangent  distance  VA  by  eq.  (21), 
and  lay  off  from  p  the  distance 
pA  =  VA  —  Vp  to  locate  the  point 
of  curve.  The  point  p  being  as- 
sumed at  random,  Vp  may  exceed  VA,  in  which  case  the  differ- 
ence pA  is  to  be  laid  off  toward  V. 

In  case  obstacles  prevent  the  direct  alignment  of  any  line 
pq,  a  line  of  several  courses  may  be  substituted  for  it  (as 


FIG.  16. 


70 


FIELD   ENGINEERING. 


explained  in  §§  46,  47,  48,)  from  which  the  length  of  pq  will 
be  deduced.  The  algebraic  sum  of  the  several  deflections  will 
equal  A. 

128.  To  locate  a  curve  wlien  the  point  of  curve  is 
inaccessible.  Fig.  17. 

Assume  any  distance  Ap  on  the  curve  which  will  reach  to 
an  accessible  point  p.  Then  by  eq.  (19)  the  angle 


Ap'  —  R  sin  pOA 
p'p  =  R  ve 
Vp'  =  VA  -  Ap' 

Measure  Vp'  and  p'p  to  locate  a  transit  point  at  p;  and  meas- 
ure an  equal  offset  from  some  transit  point  on  the  tangent,  as 
qq'.  This  gives  a  line  pq',  parallel 
to  the  tangent,  from  which  deflect  at 
p  an  angle  equal  to  pOA  for  the 
direction  of  a  tangent  through  the 
point  p. 

Instead  of  measuring  the  second 
offset  qq'  wre  may  deflect  from  pq  an 

angle  found  by  tan  qpq'  =  -^-~  and  so 

obtain  the  line  pq'  parallel  to  the 
FIG.  17.  tangent.    Or  we  may  deflect  from  p  V 

the  angle  found  by  tan  p  Vp'  =  -^—,  to  obtain  the  line  q'p  pro- 
duced, from  which  the  tangent  to  the  curve  at  p  is  found  as 
above. 

Again,  we  may  lay  off  from  V,  the  external  distance  Vh 
found  by  eq.  (24)  or  Tab.  VI  on  a  line  bisecting  the  angle 
A  VB.  This  gives  us  li,  the  middle  point  of  the  curve,  and  a 
line  at  right  angles  to  li  V  is  tangent  to  the  curve  at  U,  from 
which  the  curve  may  be  located  in  either  direction. 

129.  To  locate  a  curve  wJien  both  the  Vertex  and  Point 
of  curve  are  inaccessible.  Fig.  18. 

From  any  point  p  on  the  tangent  run  a  line  pq'  to  the  other 


SIMPLE   CURVES. 


71 


tangent,  and  so  determine  pA  as  in  §  127.     Suppose  the  curve 
produced  backward  to  p'  on  the  perpendicular  offset  pp'. 
Then 

sin  p'  OA  =  ^-  and  pp'  =  R  vers  p'  OA 

Having  located   the  point  p',  a  parallel  chord  p'q  may  be 
laid  off,  giving  a  point  q  on  the  curve,  since  p'q  =  2  X  pA. 
At  q  deflect  from  qp'  an  angle  equal  to  p'  OA  for  a  tangent  to 
the  curve  at  q. 
If  any  obstacle  prevents  using  the  chord  p'q,  any  other 


FIG.  18. 


Fia.  19. 


chord  as  p's  may  be  used,  by  deflecting  from  p'q  the  angle 
qp's  =  $  (qOs)  and  laying  off  its  length, 

p's  =  2R  sin  (p'OA  +  qp's). 

At  s  a  deflection  from  the  chord  sp'  of  (p'OA  +  qp's) -will  give 
the  tangent  at  s. 

If  obstacles  prevent  the  use  of  any  chord,  the  methods  de- 
scribed in  §  131  may  be  resorted  to. 

13O.  To  pass  from  a  curve  to  the  forward  tangent  when  the 
Point  of  Tangent  is  inaccessible.  Fig.  19. 

From  any  transit  point  p  on  the  curve,  near  the  end  of  the 
curve,  run  a  chord  parallel  to  the  tangent.  The  middle  point 
g  of  the  chord  will  be  on  the  radius  through  the  point  of  tan- 
gent B.  At  any  convenient  point  beyond  this  an  offset  equal 
lo  pp  =  R  vers  pOB  may  be  made  to  the  tangent,  and  at 
some  other  point  an  equal  offset  will  fix  the  direction  of  the 
tangent. 


FIELD   ENGINEERING. 


Otherwise,  if  an  unobstructed  line  pq  can  be  found  inter- 
secting the  tangent  at  a  reasonable  distance  from  B,  measure 
the  angle  q'pq  =  pqp',  and  lay  off  the  distance 


pg'  = 


pp' 


sin  qpq 
to  fix  the  point  q.     Then 

Bq=p'q  —  p'B  =  pp'  cot  q'pq  —  R  sin  pOB. 

Otherwise ;  assume  an  arc  of  any  number  of  stations  from 
p  to  q"  on  the  curve  produced,  and  take  the  length  of  chord 
from  Tab.  VII.  Lay  off  pq",  and  from  q"  lay  off  q"q  =  H 
vers  q"OB,  perpendicular  to  the  tangent,  to  locate  g.  The 
angle  pq"q  =  90D  —  q'pq",  and  the  distance  qB  —  E  sin  q"OB. 

131.  To  pass  an  obstacle  on  a  curve.  Fig.  20. 
From  any  transit  point  A'  on  the  curve  take  the  direction 
of  a  long  chord  which  will  miss  the  obstacle,  as  A'B' .  The 
length  of  this  chord  is  2JR  sin 
V'A'B',  V'A  being  tangent  to  the 
curve  at  A'  (see  eq.  22),  and  by 
measuring  this  distance,  the  point 
B'  on  the  curve  is  obtained.  If 
the  angle  V'A  'B'  is  made  equal  to 
the  deflection  for  an  exact  number 
of  stations,  the  chord  may  be  taken 
from  Tab.  VII. 

If  the  chord  which  will  clear  the 
obstacles  would  be  too  long  for  con- 
venience, as  A'q',  we  may  measure 
a  part  of  it  as  A'p',  and  then,  by  an 

ordinate  to  some  station,  regain  the  curve  at  p.  The  distance 
on  the  curve  from  A'  to  p  being  assumed,  the  distances  A'p' 
and  p'p  are  calculated  by  the  methods  given  in  §  121  to  §  125. 
If  p'p  can  be  made  a  middle  ordinate  the  work  will  be  much 
simplified.  If  more  convenient  the  middle  ordinate  may  first 
be  laid  off  from  A '  to  p",  and  the  half  chord  afterwards 
measured  from  p"  to  locate  p. 

Again,  we  may  calculate  the  auxiliary  tangent  A '  V  for 
any  assumed  length  of  curve  A  'B',  and  lay  off  the  distance 
A'V  and  V'B1,  deflecting  at  V  an  angle  equal  to  twice 


FIG.  20. 


SIMPLE   CURVES. 


73 


V'A'B'.  But  if  the  point  V  should  prore  inaccessible,  we 
may  conceive  the  auxilliary  tangents  to  be  revolved  about  the 
chord  A B'  as  an  axis,  so  that  V  will  fall  at  V,  and  the 
lines  A'V  and  V"B'  may  be  laid  out  accordingly.  If  these 
in  turn  meet  obstructions,  we  may  run  a  curve  from  A '  to  B' 
of  same  radius  as  the  given  curve,  but  tangent  to  A'V"  and 
V"B'. 

Again,  the  entire  curve  or  any  portion  of  it  may  be  laid  out 
by  offsets  from  the  tangents,  or  by  ordinates  from  a  long 
chord,  as  already  explained,  §  119  to  §  126. 

In  case  any  distance  on  a  curve  must  be  measured  by  a  tri- 
angulation,  as  in  crossing  a  stream,  a  long  chord  may  be 
chosen,  either  end  of  which  is  accessible,  and  the  triangula- 
tion  is  then  performed  with  respect  to  this  chord  or  a  part  of 
it,  as  upon  any  other  straight  line. 

SPECIAL    PROBLEMS   IN   SIMPLE   CURVES. 

132.  Given:  a  curve  joining  two  tangents,  to  find  the  change 
required  in  the  radius  B,  and  external  distance  E,  for  an 
assumed  change  in  the  value  of  the  tangent  distance  T.  Fig.  21. 


FIG.  21. 


Let  T  =  AV=  VB 


=  A'V=  VB' 


Then  T  —  T'  —  AA  =  the  given  change. 

Byeq.  (25)  £   =  T  cot^A 

-K'=  T  cot^A 

OG  -  R  —  R '  r=  (T  -  T1}  cot  \  A 


(48) 


74  FIELD   EHGIKEERLtfG. 

By  eq.  (26),  similarly, 

HH'  =  E  -  E'  =  (T  -  T')  tan  £  A  (49) 

Eqs.  (48)  (49)  give  the  changes  in  R  and  E  for  any  change 
in  T.  When  Tis  increased  R  and  E  will  be  increased  also, 
and  vice  versa. 

Example. — A  4°  curve  joins  two  tangents,  making  an  angle 
of  38°  =  A ,  and  it  is  necessary  to  shorten  the  last  tangent  dis- 
tance 80  feet.  What  will  be  the  change  in  the  radius  and  in 
the  external  distance? 

Eq.  (48)  T—T'  =  80  log  1.903090 

iA  19°          log  cot  0.463028 


Ans.  R   -R'    232.34  log.  2.366118 

R  1432.69 


R '  =         1200. 35      or  about  4°  46'  =  D '. 

If  the  tangent  distance  had  been  increased  80  feet  we  should 
add  the  above  to  R. 

•R'  =  1665.03        or  about  3°  26'  =  I)  ' 

Eq.  (49)        T-  T'  =  80  log  1.903090 

iA  9°  30'       log  tan  9.223607 


Ans.          E—E'      13.387  log  1.126697 

133.  Given:  a  curve  joining  two  tangents,  to  find  the  change 
required  in  the  radius  K,  and  tangent  distance  T,  for  any 
assumed  change  in  the  value  of  the  external  distance  E.  Fig.  21. 

We  suppose  HH  '  given  to  find  OG  and  A  A  '. 

By  eq.  (24)  E  =R   ex  sec  i  A 

E'  =  R'  ex  sec-A 


OG  =  R-R'=:~  (50) 

ex  sec  i  A 
By  eq.  (49) 

AA  =  T-  T'  =  (E-  E')  cot  £A  (51) 


SIMPLE   CURVES.  75 

Example. — A  4°  curve  joins  two  tangents,  making  an  angle 
of  38°  =  A ,  and  it  is  necessary  to  bring  the  middle  point  of 
the  curve  25  feet  nearer  the  vertex  F.  What  changes  are  re- 
quired in  the  radius  and  point  of  curve? 

Eq.  (50)     E-E'=          25          log  1.397940 

|  A  19°        log  ex  sec  8.760578 

Am.        R-R'      433.87          log  2.637362 

R    1432.69 


R '    998. 82  or  about  5°  44'  =  D' 

Eq.  (51)    E-E'  25  log  1.397940 

iA  9°  30          log  cot  0.776393 

T-  T'        149.39  2.174333 

or  the  P.  C.  will  be  moved  toward  the  vertex  149.39  feet. 

But  if  the  point  H,  Fig.  21,  were  to  be  moved  25  feet 
further  from  the  vertex  V,  then 

R'  =  1866.56  or  about  3°  04'  =  D' 

and  the  P.O.  will  be  moved  149.39  feet  further  from  the 
vertex. 

It  is  preferable  to  assume  some  radius  from  Table  IV.  near 
the  value  of  R '  found  as  above,  and  from  this  calculate  the 
value  of  T7'  by  eq.  (21). 

134.  Given:  a  curve  joining  two  tangents,  to  find  the  change 
made  in  the  tangent  distance  T,  and  external  distance  E,  by 
any  assumed  change  in  the  value  of  the  radius  R.    Fig.  21. 

By  eq.  (48) 

AA'  =  T-  T'  =  (R  -R')  tan  £A  (52) 

By  eq.  (50) 

HE' =  E-E'  =(R- R')exseclA  (53) 

The  changes  calculated  by  eqs.  (52)  (53)  will  be  added  to  or 
subtracted  from  T  and  E  respectively,  according  as  the  radius 
is  increased  or  diminished. 

135.  Since  for  a  constant  value  of  the  central  angle  A, 


76 


FIELD   ENGINEERING. 


the  homologous  parts  of  any  two  curves  are  proportional  to 
each  other,  we  may  write  at  once 


0' 


(54) 


etc.  etc.  etc. 

136.  Given:  a  curve  joining  two  tangents,  to  change  the 
position  of  the  Point  of  curve  so  that  the  curve  may  end 
in  a  parallel  tangent.  Fig.  22. 

Let  AB  be  the  given  curve,  AV,  VB  the  tangents,  and 
V'B '  the  parallel  tangent.  Then  W  is  the  distance  from 
one  vertex  to  the  other;  and  since 
there  is  no  change  in  the  form  or 
dimensions  of  the  t;urve,  we  may 
conceive  it  to  be  moved  bodily, 
parallel  to  the  line  AV,  until  it 
touches  the  line  V'B',  when  every 
point  of  the  curve  will  have  moved 
a  distance  equal  to  VV.  Hence 
AA  '  =  00 '  =  BB '  =  VV.  There- 
fore, run  a  line  from  B  parallel  to 
AV,  intersecting  the  new  tangent  in 
B',  measure  BB',  and  lay  off  the  dis- 
tance from  A  to  find  A.  In  the  figure  the  new  tangent  is 
taken  outside  the  curve,  and  so  A '  falls  beyond  A,  but  if  the 
new  tangent  were  taken  inside  the  curve  at  V"B",  the  new 
P.  C.  would  fall  back  of  A  at  some  point  A". 

If  the  parallel  tangent  is  defined  by  a  perpendicular  offset 
from  B,  as  Bp;  since  the  angle  BB'p  =  A 


FIG.  22. 


sin  A 


(55) 


137.  Given:  a  curve  joining  two  tangents,  to  find  the 
radius  of  a  curve  that,  from  the  same  Point  of  curve,  will  end 
in  a  parallel  tangent.  Fig.  23. 

Let  AB  be  the  given  curve,  AV,  VB  the  tangents,  and 
V'B'  the  parallel  tangent;  and  let  AO  =  R  and  AO'  =  R'. 


SIMPLE   CURVES.  77 

Since  the  central  angle  A  remains  unchanged,  the  angle 
•JA  between  the  tangent  and  long  chord  remains  unchanged; 
therefore  V A  B '  =  VAB,  and  the  new  point  of  tangent  is  on 
the  long  chord  AB  produced.  Find  on  the  ground  the  inter- 
section of  V'B'  with  AB  produced 
and  measure  BB'.  In  the  diagram 
draw  Be  parallel  to  AO,  then  BeB '  = 
A,  and  by  eq.  (22) 

but 

Be=  00'  =  R'  -  R 

...         R>  =  R  +  J^—        (56) 
'2  sin -I- A 

. 

The  -f-  sign  is  used  when  B '  is  be-  FlG-  23- 

yond  B,  as  in  the  figure;   but  if  the 

parallel  tangent  is  within  the  given  curve  it  will  cut  the 
chord  in  some  point  B ",  and  then  the  —  sign  must  be  used, 
since  R '  will  evidently  be  less  than  R. 

If  the  parallel  tangent  is  defined  by  a  perpendicular  offset, 
as  Bp  =  B  'f;  since  BeB '  =  A 

Bp  —  Be  vers  A  =  (R '  —  M)  vers  A 

R'  =  RA ?2—  (57) 

'   vers  A 

Add  or  subtract  as  explained  above. 

If  the  long  chord  C  =  AB  is  known,  then  the  new  long 
chord  C'  =  AB '  or  AB"  =  C  ±  BB',  and  by  eq.  (54) 


138.  Given:  a  curve  joining  two  tangents,  to  change  the 
radius,  and  also  the  Point  of  curve,  so  that  the  new  curve 
may  end  in  a  parallel  tangent  directly  opposite 

the  given  Point  of  tangent.     Fig.  24. 

Let  AB  be  the  given  curve,  A  V,  VB  the  tangents,  V'B'  the 
parallel  tangent,  and  B '  the  given  tangent  point  on  the  radius 
OB  produced. 


78 


FIELD   ENGINEERING. 


In  the  diagram,  produce  the  tangent  .AFand  the  radius  OB 
to  intersect  at  K.    Then 

BK  =  R  exsec  A 
B  '  K  =  R  '  exsec  A 
Subtracting  we  have 

BB'  =  (R-R')  exsec  A 


exsec  A 
from  which  R'  is  easily  determined,  as  in  §§  132  and  133. 


(59) 


FIG.  24. 


FIG.  25. 


To  find  the  change  A  A'  of  the  P.O.,  in  the  diagram  draw 
O'G  parallel  to  A' A;  then 


or 


0'G  =  OG  tan  A 
AA  =  (R  —  R')  tan  A 


(60) 


By  substituting  the  value  of  (R  —  R')  from  eq.  (59)  and  ob- 
serving Table  II.  42  we  have 


AA  =BB'  X  cot-l-A 


(61) 


Observe  that  eqs.  (59),  (60),  and  (61)  may  be  derived  directly 
from  eqs.  (50),  (52),  and  (51)  respectively  by  writing  A  for  i  A. 

139.  Given:  a  curve  joining  two  tangents;  to  find  the  new 
tangent  points  after  each  tangent  Ms  been  moved 
parallel  to  itself  any  distance  in  either  direction.  Fig.  25. 


SIMPLE   CURVES.  70 

Let  A  and  B  be  the  given  tangent  points,  and  A'  and  B  ' 
the  new  tangent  points  required.  Let  the  known  perpendicu- 
lar distances  Aq  =  a,  and  Bp  =  b.  We  then  require  the 
unknown  parallel  distances  q  A  =  x  and  pB  '  =  y. 

Since  the  form  and  dimensions  of  the  curve  remain  un- 
changed we  may  conceive  the  curve  to  be  moved  bodily 
into  its  new  position  on  lines  parallel  and  equal  to  the 
line  VV  joining  the  vertices.  Then  A  A  =  00'  =  BB1  = 
VV. 

In  the  diagram  draw  VK  parallel  and  equal  to  Bp  =  b  and 
VH  parallel  and  equal  to  Aq  =  a.  Then  VH=  qA  =  x,  and 
V'K=  B'p  =  y.  Since  VG  V  =  A,  we  have 


VG  =  -T-—  and  GH=  —^ 

sin  A  tan  A 

and  since 

VH=  VG-GH=x 


b 


sin  A        tan  A 
Similarly  (62) 

b  a 

nj   —   

~  tan  A       sin  A 


When  the  new  tangents  are  outside  of  the  given  curve,  the 
offsets  a  and  b  are  considered  positive;  if  either  new  tangent 
were  inside  of  the  given  curve  its 
offset  would  be  considered  negative. 
In  solving  eqs.  (62)  if  a-  and  y  are 
found  to  be  positive  they  are  to  be 
laid  off  forwards  from  q  and  p,  as 
in  Fig.  25;  if  either  is  found  to  be 
negative  it  is  to  be  laid  off  in  the 
opposite  direction. 

Example. — A  certain  curve  has  a 
central  angle  of  50°  —  A ,  and  it  is 
proposed  to  move  the  first  tangent 
in  20  feet  and  the  second  tangent 

out  12  feet.     Required,  the  distances  on  the  tangents  from  the 
old  tangent  points  to  the  new.     Fig.  26. 


80  FIELD   EHGIHEEKI^G. 

Here  a  =  -  20  and  b  =  -f  12 


+  b 

12 

1.079181 

-a  20 

1.301030 

A 

50" 

log  sin 

9.884254 

A   50° 

log  tan 

0.076186 

15.665 

1.194927 

-   16.782 

,1 

1.224844 

x  =  15.665  -(-  16.782)  =  +  32 

450 

+  & 

12 

1.079181 

-a  20 

1.301030 

A 

50° 

log  tan 

0.076186 

A    50° 

log  sin 

9.884254 

10.069 

1.002995 

'ii''i  26.108 

1.416776 

y  =  10.069  -  (-  26.108)  =  +36.177 

i  x  =  -  32.450 
For  -J-  a  and  -b  \y=-  36.177 


x-  -    1.120 
=-  15.989 


<«  =  +   1.120 
For  -  a  and  -  b  \y  =  +  15.939 

If  we  have  a  and  x  given  to  find  b  and  y:  Solving  eqs.  (62) 
for  b  and  y  we  obtain 


=  x  sin  A  -*••'«  cos  A  |  //»ox 


y  =  «  cos  A  —  «•  sn  A 


In  which  the  algebraic  signs  of  the  quantities  must  be  ob- 
served as  above. 

14O.  Given:  a  curve  joining  two  tangents,  to  find  a  new 
Radius  and  new  position  of  the  Point  of  curve,  such 
that  the  curve  may  end  at  the  same  point  as  before,  but  with 
a  given  change  in  the  direction  of  the  forward  tangent. 
Fig.  27. 

Let  AB  be  the  given  curve,  AV,  VB  the  given  tangents, 
V'B  the  new  tangent,  and  VBV  the  given  change  in  direc- 
tion. Let  A'  =  A  +  VBV. 


SIMPLE   CURVES. 


81 


In  the  diagram  draw.Btr  perpendicular  to  AV produced; 
then 

BG  =  R  vers  A 

=  R'  vers  A' 
Hence 


vers  A 


(64) 


and 


AA  =  AG  —  A'G  =  Rsm  A  —  R'  sin  A'        (65) 


In  the  figure  the  change  in  direction  of  tangent  makes  A' 
greater  than  A ;  therefore  V  falls  beyond  F,  and  A  beyond 


.  27. 


FIG.  28. 


A  ;  but  if  the  change  made  A  '  less  than  A  ,  then  V  and  A 
would  fall  behind  V  and  A  respectively,  and  R  '  would  be 
greater  than  R. 

The  same  formulae  apply  to  the  converse  problem  in  which 
B  is  taken  as  the  point  of  curve,  and  A  and  A  as  points  of 
tangent. 

141.  Given  a  curve  joining  two  tangents,  to  find  the  change 
in  the  Point  of  curve  when  the  forward  tangent  takes  a  new 
direction  from  the  vertex  V.  Fig.  28. 

By  eq.  (21) 


AA'  =  R  (tan  i  A  —  tan  £  A^)  (66) 

142.  Given:  a  curve  joining  two  tangents,  to  find  the  new 


OZ  FIELD    EKGIKEERING. 

radius,  R',  wlien  the  forward  tangent  takes  a  new  direc- 
tion from  the  vertex,  V.    Fig.  29. 
By  eqs.  (21)  (25) 


cotiA' 


(67) 


143.  Given:  a  curve  joining  two  tangents,  and  a  given 
change  in  the  direction  of  the  forward  tangent  from  the 
vertex,  to  find  the  radius  and  point  of  curve  of  a  curve 
that  shall  pass  at  the  same  distance,  VH,  from  th^  vertex. 
Fig.  30. 

Let  AS  be  the  given  curve,  BVB'  the  given  change  in 


FIG.  29. 


FIG.  30. 


direction  of  tangent,  and  VH'  =  VH.    Let  A'  =  A  +  BVB'-, 
then  eq.  (24) 

VH=  -Rex  sec  4 A  —  VH'  =  R'  ex  sec  iA' 


By  eq.  (28) 


exsec 


(68) 


FA=F5"cotiA,     VA  =  FIT  cot  i  A' 

AA  =  FH"(cot  iA  —  cot  £A')  (69) 

But  in.  case  A'  =  A  —BVB',  A  A  becomes  negative  and 
must  be  laid  off  backward  from  A. 


SIMPLE   CURVES. 


83 


Example.— Given  a  2°  curve,  A  =  80°  and  BVB'  =  -  10' 

.-.  A'  =  70° 


1A 

VH 

IA' 

JR' 


40° 

874.97 
35° 

1°  27'  nearly 
20° 
17°  30' 


log  3.457114 
log  exsec  9.484879 

2.941993 
log  exsec  9.343949 


3.598044 


cot  2.74748 
cot  3.17159 

-  0.42411 


AA  =  874.97  X  (-  .42411)  =  -  371.08 
and  must  be  laid  off  backward  from  A. 

144.  Given:  two  indefinite  tangents,  a  point  situated  be- 
tween them,  and  tJie  angle  A,  to  find  the  radius  R,  and  tan- 
gent distance  T  of  a  curve  joining  the  tangents  which  shall  pass 
through  the  given  point.  Fig.  31. 

If  the  given  point  is  on  the  bisecting  line  VO,  as  H,  meas- 
ure VH=  E,  and  find  R  and  Tas  in  §§  97,  98. 
When  the  given  point,  as  P  is  not  on  the  bisecting  line  VO; 
if  a  line  GK  is  passed  through  P  per- 
pendicular to  VO,  it  will  be  parallel 
to  any  long  chord,  as  AB,  and  the 
angle  VGK—%&.  The  curve  pass- 
ing through  P  will  intersect  GK  in 
some  other  point  P' ;  the  line  GK 
is  bisected  by  the  line  VO  at  /,  and 
PI=  P'L 

If  the  given  point  P  is  located  by  a 
perpendicular  off  set  from  the  tangent, 
asPZ;  in  the  triangle ' PLG,  LG  — 
PL  cot  | A.  Lay  off  LG,  and  at  G  deflect  VGK=  |A,  and 
measure  GP  and  PK.  Since  by  Geom.  (Tab.  I.  24)  GA*  = 
GP'  X  OP,  and  OP'  -  PK; 


FIG.  81. 


OA=  V 


(70) 


84 


FIELD   ENGINEERING. 


Lay  off  GA;  and  A  is  the  Point  of  curve,  AV=  T,  and 
R  =  AVcotiA. 

If  the  given  point  were  located  by  an  offset  from  BV,  find 
B  first,  and  make  VA  =  BV. 

If  the  given  point  Pis  located  by  a  perpendicular  offset 
IP  from  the  bisecting  line  VO;  produce  IP  to  intersect  the 
tangent  at  G  and  measure  PG.  Since  P'G  =  GP  -\-  2PI 


GA=  VGP(GP+2PI)  (71) 

whence  we  have  the  point  of  curve  A,  as  before. 

145.  Given:  a  curve,  AP,  and  the  radial   offset  PP' 

to  Jind  a  curve  which  shall  pass  through  the  point  P ',  start- 
ing from  the  same  point  of  curve  A.    Fig.  32. 


Let  b  —  PP',  and  in  the  diagram  draw  P'G '  parallel  to  the 
common  tangent  AX,  and  join  AP'.     Then 

P'G'  =  (R  ±  ft)  sin  A 
G'A   =  R  -  (R  ±  b)  cos  A 


tan  4-  A  ' _ 

P'G'~  (R  ±  b)  sin  A 


—  cot  A 


R'  = 


(JS  ±  b)  sin  A 


sn  A 


sin  A 


(72) 


(73) 


When  the  offset  is  outward  use  R  +  &»  when  it  is  inward 
use  R  -  b. 

Example. — Given:  a  3°  curve  of  16  stations  and  a  radial 
offset  of  205  feet  inward  from  the  P.  T.  to  find  the  radius  of 
the  curve  passing  through  the  extremity  of  the  offset. 


SIMPLE   CURVES.  85 


Here  A  =  3°  X  16  =  48°;  and  b  =  205. 

H    3°  =     1910.08 

R-b         1705.08  log  3.231745 

A     48°  log  sin  9.871073 

P'G'  3.102818 

RZ°  log  3.281051 

1.50742  0.178233 
A     48°  cot  .90040 


iA'        tan  .60702  =  31°  15J' 

2 

A'  62°  31'  log  sin  9.947995 

P'G'  log  3.102818 

R'  (about  4°  01').     Ans.  3.154823 

If  the  same  offset  were  made  outside  of  the  curve  we  should 
find  R1  log  3.438350,  or  about  a  2°  05'  curve. 

This  solution  is  inconveniently  long  for  ordinary  field  prac- 
tice. When  the  offset  is  small  compared  with  the  length  of 
curve,  we  may  use  the  following 

Approximate  Rule :  Divide  twice  the  offset  6  by  the 
length  of  curve,  look  for  the  quotient  in  the  table  of  nat. 
sines,  and  take  out  the  corresponding  angle,  which  multiply  by 
100,  and  divide  by  the  length  of  curve.  The  quotient  is  the 
correction  for  the  given  degree  of  curve ;  to  be  subtracted  when 
the  offset  is  made  outward,  and  added  when  the  offset  is  made 
inward. 

This  rule  is  expressed  by  the  formula 

J-  =  .DT™  *„-'£.  (74) 

Ju  Li 

Taking  the  same  example,  we  have 
~  =  sin  14°  51' 

Ju 

and  correction  =  14°  51'  X  ;  ^r-  =  T  0°  56' 
1600 

Hence  D'  —  3°  56'  or  D'  =  2°  04' 


86  FIELD    ENGINEERING. 

THE    VALVOID. 

14G.  Given:  any  number  of  circular  curves  of  equal  length 
L,  all  starting  from  a  common  point  of  curve  A,  in  a  common 
tangent  AX,  to  find  the  equation  of  the  curve  joining 
their  extremities.     Fig.  33. 
Let  AP  be  any  one  of  the  given  curves, 
"   R  =  its  radius  AO, 
"   D  =  its  degree  of  curve, 
"    A  =  its  central  angle  AOP, 
"    C  =  its  Ion?  chord  AP. 


FIG.  33. 

By  substituting  the  value  of  ft  from  eq.  (16)  in  eq.  (23)  we 
have 

C=l<X)!$li  (75) 


Substituting  in  this  the  value  of  D  from  eq.  (20)  and  letting 
heta)  fl  =  -J  A,  (rho)  p  —  j^-  and  J 

1UU 

polar  equation  of  the  required  curve 


(theta)  fl  =  -J  A,  (rho)  p  —  j^-  and  JV  =  \~,  we  have  for  the 

1UU  1UU 


(76, 


in  which  p  is  the  radius-  vector  AP,  0  the  variable  angle 
XAP,  the  unit  of  measure  is  one  side  of  the  inscribed  polygon 
by  which  the  circular  curve  AP  is  measured,  and  N  the  num 
ber  of  these  sides  in  the  length  of  the  curve  AP.  By  the 


SIMPLE   CURVES.  87 

conditions  of  the  problem  N  is  constant,  but  6  may  have  any 
value  whatever.  If  we  let  0  vary  from  0°  to  -j-  180°  and  from 
0°  to  —  180°  the  point  X  will  describe  the  curve  XP'PA 
shown  in  the  figure,  which  is  called  the  Valvoid  from  its  re- 
semblance to  the  shell  of  a  bivalve.  All  circular  curves  tan- 
gent to  AX  at  A  and  having  a  length  L  =  AX  will  terminate 
in  the  valvoid,  and  the  line  PP'  joining  the  extremities  of 
any  two  of  them  is  a  chord  of  the  valvoid. 

147.  To  find  a  tangent  to  the  valvoid  at  any  point 

JP.     Fig.  34.     See  Appendix. 
Differentiating  eq.  (76) 

'  (77) 

which  is  essentially  negative,  since  p  is  a  decreasing  function 
of  0. 

Let  (phi)  cp  =  APG,  the  angle  between  the  radius  vector 
and  the  normal  PG. 

tan  cp  —  — r  cot  — -  —  cot  0  (78) 

The  line  PR  perpendicular  to  PG  is  tangent  to  the  valvoid 
at  P,  and  PV  perpendicular  to  PO  is  tangent  to  the  curve  AP. 

Then  APV  =  0  and  VPG  —  6  —  q>,  and  letting  i  =  OPK  = 
VPG. 

i=Q -q>  =  lA-q>  (79) 

Therefore,  to  obtain  the  direction  of  a  tangent  to  the  val- 
void at  any  point  P,  deflect  from 
the  radius  PO  an  angle  equal  to 
z '  =  (£  A  —  (p),  on  the  side  of  PO 
farthest  from  the  point  ot  curve  A. 

The  value  of  i  may  be  found  by 
eqs.  (78)  (79),  but  we  are  saved 
this  somewhat  tedious  calculation 
by  the  use  of  Table  X.  1,  which 

contains  values  of  the  ratio  —  =  u  Fl°-  34< 

A 

for  various  values  of  A,  and  length  of  curve  L.  Multiplying 
A  by  the  proper  tabulated  number  gives  the  value  of  i  —  OPK 
at  once ;  or 

»  =  (iA  -<?)  =  u  A  (80) 


88  FIELD 


148.  To  find  the  radius  of  curvature  of  the  valvoid 

at  any  point  P.     See  Appendix. 
Differentiating  eq.  (77)  we  have 


The  general  formula  for  the  radius  of  curvature  of  polar 
curves  is 


r  = 


Substituting  in  this  the  values  of  p,  —-,  and  •—,  and  putting 

(-r-f  cot  —-  —  cot  6  }  =  a  we  have  after  reduction, 
J\         N  I 


This  formula  being  too  complicated  for  convenient  use  in 
the  field,  its  use  is  avoided  by  referring  to  Table  X.  2,  which 

contains  values  of  the  ratio  T  =  v  f  or  various  values  of  A  and 

Ju 

L.  Multiplying  the  given  value  of  L  by  the  proper  tabular 
ratio,  gives  the  value  of  the  radius  of  curvature  of  the  valvoid 
for  a  short  distance  either  way  from  the  given  point  P,  or, 

r  =  vL  (82) 

149.  To  find  the  length  of  arc  of  the  valvoid  corre- 
sponding to  a  change  of  one  degree  in  the  value  of  tlie 
angle  A.  Fig.  35. 

From  any  chord  AP  suppose  a  deflection  of  £  degree  to  be 
made  each  way  to  Ap'  and  Ap"  ;  then  the  angle  p'Ap"  =  -£°  = 
the  change  in  0,  and  since  A  =  20,  this  makes  a  change  of  1° 
in  the  value  of  A.  We  then  require  to  know  the  length  of 


SIMPLE   CURVES. 


89 


the  arc  p'p",  and  we  may,  without  sensible  error,  consider  it 
to  be  described  by  the  radius  of  curvature  r  =  Po  for  the 
point  P,  through  an  angle  p'op".  Now 


(A  '                      \               /    A  *  \ 

— — h  <?>'  1  —  |-x 95" )  = 
6                      I             \    A  I 

JrL.  _  A!  _i_     '  _     * 
By  eq.  (80) 


—  2u')       and 


A  " 

— - 


Fia  .33. 

and  since  <p'  is  so  nearly  equal  to  <p*  we  may  assume  u1  = 
U"  =  u  ;    hence    q>  —  q>*  =  -  - — (1  —  2u)  and  p'op*  = 

(A'-  A')(l-tO; 

But  the  condition  of  the  problem  requires  A'  —  Aff  =  l°, 
hence  p'op"  =  (1  —  u)°. 

Therefore  the  length  of  arc  p'p"  for  a  change  of  1°  in  the 
value  of  A  is 

lt  =  r(l  —  u)  X  arc  1° 

or  (Tab.  XVII.)        I,  =  r  (I  —  u)  .0174533 
and  since  r  =  vL  (Tab.  X.  2), 

l,  =  i)(\.  —  u)L  .0174533  (83) 

By  this  formula  Table  X.  3  has  been  prepared,  for  various 
values  of  A  and  L. 

15O.    Given:  two  curves  of  the  same  length  L  but  of 
different  radii,  starting  from  the  same   point  of  curve   in  a 


90 


FIELD   EKGINEEKING. 


common  tangent,  to  determine  the  direction  and  length  of 
a  line  joining  their  extremities.    Fig.  36. 

Let  AX  be  the  common  tangent,  and  AP ',  AP"  the  two 
curves,  to  determine  the  direction  and  length  of  PP". 

If  we  take  the  point  P  on  the 
arc    P'P"    determined    by  the 

A'  +  A 

angle    A  = £ and  draw 

A 

a  tangent  PK  to  the  valvoid  at 
P,  we  may  assume  without  ma- 
terial error  that  the  chord  P'P" 
will  be  parallel  to  PK  for  any 
value  of  P'P"  not  exceeding 
\L,  a  limit  not  likely  to  be  ex 
ceeded  in  practice. 
Let  0  be  the  centre  of  the  curve  AP  fixing  the  point  P ; 


then  AOP  = 


— ,  and 


Since  P'P"  is  assumed  parallel  to  PK, 


PP'O"  =  KGO"  =  A"-  K=  A"- 
P'P'O"  =  i"=  A'fl  +  ^l 


A'" 


2 


Similarly  producing  P"P  to  any  point  IT, 


HFO'  =  t  = 


whence  also 


f  =  i*  +  A'  -  A" 


(1-u) 


(84) 

(85) 
(85)' 


The  slight  error  involved  in  the  above  assumption  is  cor- 
rected by  taking  out  the  value  of  u  (Table  X.  1)  correspond- 
ing t.o  A",  the  less  of  the  two  given  central  angles;  we  have 
therefore  written  u  with  the  double  accent  in  equations  (84) 
and  (85). 


SIMPLE   CUKVES.  01 

When  i'  and  i*  are  positive,  they  will  be  deflected  as  in 
Fig.  36,  on  the  side  of  the  radius  farthest  from  A  ;  should  i"  be 
negative  it  will  of  course  be  deflected  from  P"0"  toward  A. 

The  arc  P'P"  corresponds  to  a  change  of  the  central  angle 
from  A'  to  A"  ;  hence 

1°  :   A'-  A"  ::  I,  :  P'P" 
or 

P'P"  =  .(A'-  A"K  (86) 

in  which  I,  is  taken  from  Table  X.  3  for   L  =  AP,  and 


As  in  practice,  the  distance  P'P"  is  usually  small  compared 
with  L,  the  arc  and  chord  will  be  almost  identical  and  no 
further  calculation  is  necessary.  If  P'P"  is  large,  it  will  be 
found  that  equation  (86)  gives  the  length  of  arc  very  correctly 

when  --  —  <r  -  does  not  exceed  20°,  and  the  length  of  chord 

when  —  ^  -  exceeds  60°  ;  for  intermediate  mean  angles  it 
a 

gives  a  value  to  P'P"  between  that  of  the  arc  and  chord. 
The  arc  P'P"  may  be  considered  to  be  described  by  the  radius 

r  —  vL,  v  being  taken  for  —  ~—  -  (Table  X.  2),  and  its  total 

curvature  is  found  by  multiplying  its  length  by  the  degree  of 
curve  corresponding  to  r  (Table  IV). 

Example.  Given,  a  2°  30'  curve,  and  a  1°  curve  of  12  stations 
each  from  the  same  PC,  to  determine  the  distance  between 
their  extremities. 


A'  =  2i°  X  12  =  30°,         A"  =  12°,        —    r      -  21°, 

a 

A'-  A"  =  18°,  uv  =  .33446 

Eq.  (84).   i"  =  2°.9737  =    2°58'25" 

Eq.  (85)'.  »'  =  *'"  +  A'  -  A"  =  20°.9737  =  20°58'25" 
Eq.  (86).   Arc  P'P"  =  18°  X  10.425          =  187.65  ft.    Ans. 
Eq.  (82).   r  =  1200  X  .7479  =  897.48  ft.  =  (say)  a  6°23'  curve. 
Total  curvature,  P'P"  =  6°.  383  X  1.8765  =  11°.  9777. 
(The  distance  P'P"  may  be  found  by  solving  the  triangle 

formed  by  itself  and  the  long  chords  of  the  curves  AP't 

AP".) 


92  FIELD 


151.  Given:  a  curve  AP,  to  find  a  curve  starling  from  the 
same  point  A,  that  shall  shift  the  station  P  any  desired  dis- 
tance PP'  to  the  right  or  left.  Fig.  36. 

Before  we  can  determine  what  distance  PP'  is  desired,  we 
must  know  (approximately)  its  direction.  We  have  given, 
therefore,  D,  L,  and  A  to  find  the  angle  OPP',  and  (after 
measuring  PP')  to  find  A'  and  D  '. 

The  solution  is  necessarily  somewhat  approximate,  yet 
close  enough  for  all  practical  purposes.  For  if  the  required 
value  of  D'  were  obtained  precisely,  it  would  probably  involve 
some  seconds,  and  would  therefore  be  discarded  in  favor  of 
some  value  in  even  minutes. 

When  P'  is  inside  the  given  curve  : 

Eq.  (80).  i  =  OPK  =  UA.     Table  X.  1. 

Eq.  (82).  r  =Po       =  vL.       Table  X.  2. 

Let  d  (delta)  =  degree  of  curve  corresponding  to  r,  by 
Table  IV. 

OPP'  =  »•  _  -  i*  nearly. 


PP' 

Eq.  (86).  A'  =  A  +  —  .     Table  X.  3. 

Instead  of  taking  lt  from  Table  X.  3  for  the  exact  value 
of  A  it  is  well  to  take  it  for  the  estimated  value  of  —       —  . 


Eq.  (20).  D'  =  ^  A' 

When  P'  is  outside  of  the  given  curve  : 
i  =  u  A  ,        r  =  vL, 

180°  -  OPP'  =  t'+  £?  •  tf  nearly. 

A'  =  A  -  —         D'  =  —     ' 
Example.  Given,  a  4°  curve  of  800  feet,  or  A  =  32°  to  find 


SIMPLE   CURVES. 


93 


a  curve  from  the  same  P.  C.  which  shall  shift  the  last  station, 
in,  about  55  feet.     (Fig.  36.) 

*  =  32°  X  .3355  =  10°.736 
r  =  800  X  .7450  =  596,        .-.    S  =  9°  36'  =  9°.6 

OPP'  =  10°.736  -  -^  X  4°.8  =  8°  06' 

D'  =  ~  =  5°.     Am. 

For  a  5°  curve,  the  true  distance  PP '  =  55.53 
«   «4°>59'  «       «       «          «        PP'  =  54.60 

which  proves  this  method  practically  correct. 

152.  Given:  a  tangent  and  curve,  and  a  straight  line 
intersecting  them,  making  a  given  angle  with  tlie  tangent  at 
a  given  point,  to  determine  the  distance  on  the  line 

from  the  tangent  to  the  curve.     Fig.  37. 


FIG.  37. 

We  have  OA,  AG,  and  the  angle  AGP  to  find  OP. 
tan  AGO  =  ^-         POO  =  AGO  -  -AGP 


sin  POO 


84  FIELD   ENGINEERING. 

When  AOP  =  AGO,  cq.  (24), 

GP  =  R  exsec  (90°  -  AGO) 
When  AGP=  90°,  §§(92),  (119), 


•   R 
When  AGP'  >  AGO,  we  have 

P'GO  =  AGP'  -AGO 

but  the  other  formulae  remain  unchanged. 
Example.—  Let  R  =  955.37,  AG  =  350,  AGP=  40° 

R        955.37  log  2. 9801 70 

AG     350.  log  2.544068 


PGO 
AGO  69°  52'  47" 

OPI 
POG 

R 

PG             72.40    Ana. 

69°  52'  47" 
40° 

log  tan  0.436102 

log  sin  9.  697387 
log  sin  9.  972653 

29°  52'  47" 
32°  02'  36" 

log  sin  9.  724734 
log  sin  8.576953 

8.879566 
log  2.  9801  70 

2°  09'  49" 

log  1.859736 

This  problem  may  be  used  in  passing  from  a  tangent  to  a 
curve  when  the  tangent  point  is  obstructed.  The  distance 
^4Pon  the  curve  is  defined  by  the  angle  ^4  OP,  which  is  readily 
found. 

If  AGP'  >  2 AGO  the  line  will  not  cut  the  curve. 

153.  Given :  a  curve  and  a  distant  point  to  find  a 
tangent  that  shall  pass  through  the  point.  Fig.  38. 

We  have  the  curve  adg  and  the  point  P  visible,  but  distance 
unknown,  to  find  the  point  of  tangent  P. 


SIMPLE    CUIiVES. 


95 


Any  chord,  as  bf,  parallel  to  the  required  tangent,  if  pro- 
duced will  pass  the  point  P  at  a  perpendicular  distance  equal 
to  the  middle  ordinate  of  that  chord.  Ranging  across  every 
two  consecutive  stakes  on  the  curve  we  at  first  find  the 
range  falling  outside  of  the  required  tangent,  as  bcG,  cdll, 
•  etc. ;  but  finally  the  range  falls  inside,  as  deK.  "We  then  know 
that  the  required  point  is  between  c  and  e.  \ 

If  the  range  ce  falls  inside  the  point  P,  a 
perpendicular  distance  equal  to  the  middle 
ordinate  of  ce,  the  tangent  point  is  at  d. 
If  the  perpendicular  distance  is  greater 
than  this,  the  point  B  is  between  c  and  d. 
If  less,  or  if  the  range  ce  falls  outside  of 
P,  the  point  B  is  between  d  and  e.  The 
middle  ordinate  for  ce  (200  feet)  equals  the 
tangent  offset  for  100  feet,  given  in  Tab. 
IV.,  and  it  is  generally  so  small  that  it  can 
be  estimated  at  ^without  going  to  lay 
it  off. 

To  find  the  exact  point  B,  when  it  falls 
between  d  and  e,  find  by  trial  a  point  x 
on  the  arc  cd  in  range  with  e  and  a  point 
inside  of  P  a  perpendicular  distance  equal 
to  the  middle  ordinate  of  ex.  The  point  B 
is  at  the  middle  point  of  the  arc  ex.  If 
the  point  B  is  between  c  and  d,  stand  at  c 
and  find  a  point  x  on  the  arc  de  in  the  same 
way.  B  is  at  one  half  the  arc  ex. 

The  middle  ordinate  of  any  chord  ex  is 
less  than  M  for  200  feet,  and  greater  than  m  for  100  feet, 
necessary,  its  exact  value  m'  can  be  found  by 


Fro.  38. 


If 


,  _  m  x  ex* 
"lOOOCT 


(87) 


and  this  equation  is'  nearly  true  when  ex  is  as  great  at  300  or 
400  feet.  That  is,  middle  ordinates  on  the  same  curve  are  to 
each  other  as  the  squares  of  their  chords  very  nearly. 

By  this  method  the  point  B  is  found  without  the  use  of  the 
transit,  so  that  the  plug  can  be  driven  at  B  before  the  transit 


96 


FIELD   ENGINEERING. 


is  brought  up  from  the  rear.     It  is  therefore  preferable  to  the 
following  solution.     Fig.  39. 

From  any  two  points  a  and  c  of  the  curve  measure  the 
angles  to  the  point  P,  so  that  with  the  chord  ac  as  a  base, 
and  the  measured  angles,  we  may  find  cP  by  the  formula 

sin  caP 
cP  =  ac  — 

sin  cPa 

Knowing  the  angle  c  that  cP  makes  with  a  tangent  at  c,  we 
find  the  length  of  the  chord  cd  by  cd  =  2E  sin  c. 
By  Geom.  Tab.  I.  24, 


PB=Pe  =  VcP  X  dP 


whence  we  know  ce.    Opposite 
with  the  radius  Pe,  we  find  B. 


or  on  the  arc  eB  described 


FIG.  39. 


Fi«.  40. 


154.  Given:  two  curves  exterior  to  each  other,   to 

find  the  tangent  points  of  a  line  tangent  to  both  and  its 
length  between  tangent  points.  Fig.  40. 

Let  B  and  A  be  the  required  tangent  points.  Let  OB  =  R, 
&ndO'A  =  E'. 

On  the  curve  of  greater  radius  R  select  a  point  H  supposed 
to  be  near  the  unknown  tangent  point  B,  and  knowing  the 


SIMPLE   CURVES.  97 

direction  of  the  radius  Oil,  find  on  the  other  curve  a  point  K 
having  a  radius  0  'K  parallel  to  OH,  and  measure  HK.  In 
the  diagram  draw  Ob  and  O'a  perpendicular  to  HK.  Then 
the  angle  KO'a  =  90°  -  HKO'  =  KO'A  nearly,  which  is  the 
angle  required.  We  have  therefore  to  find  the  correction 
aO'A  =  x,  and  apply  it  to  KO'a. 

Aa  =  R'  vers  KO  'a;        Bb  —  R  vers  KO  'a  nearly. 

Ka  =  R'  sin  KO  'a  ;         Hb  =  R  sin  KO  'a 

Bb  —  Aa  =  (R—R')  vers  KO'a 

ab   =HK  -\-  (R  -  R')  sin  KO'a 

(R  -^  R')  vers  KO'a 

(88) 


KO'A  =  (KO'a  -  a-)  =  HOB 

Observe  that  KO  'a  —  the  angle  between  the  tangent  at  K  or 
H  and  the  line  IIK  ;  and  KO  'A  =  the  angle  between  the 
tangent  at  K  or  H  and  the  required  tangent  BA. 

If,  instead  of  H  and  K,  the  points  H'  and  K'  had  been. 
selected,  then 

(R-R')versH'Ob  .       /oov 

Sm  X  =  WK^(R  -  R  ')  aiOTOft  nearly>    (88) 
and 

H'OB  =  K'O'A  =  II'  Ob  -f  x. 

The  length  of  BA  should  be  obtained  by  measurement,  but 
it  may  be  calculated  by 

AB  =  ab  -  (R  -R')  sinx  (89) 

When  R  =  R',  x  —  0,  and  HKis  parallel  to  BA. 

In  case  the   curves   are  reverse   to  each  other,   as  in 
Fig.  41, 

'a 

(90) 


KO'A  =  HOB  =  KO'a  -  x 
If  the  points  H'  and  K'  are  selected,  Fig.  41, 

'  Ob 


H'OB  =  K'O'A  =  H'Ob  +  x. 


98 


FIELD   ENGIXEERIKG. 


The  lines  IIK,  AB,  and  00'  all  intersect  in  a  common 
point  J,  Fig.  41.  " 


IB  =   VHI(HI+  21t  sin  HOb) 


(92) 


(93) 


(94) 


These  last  three  equations  furnish  another  method  of 
solving  the  same  problem.  They  may  be  applied  to  Fig.  40 
by  changing  the  sign  of  R'. 

In  Fig.  41,  if  R  =  R',  then  HI—  \HK  and  AE  =  2IB. 


FIG.  41. 


Fio.  42. 


155.  Given:  two   curves,  O  and   O,  reverse   to 

each  other,  joined  by  a  tangent  BA',  and  terminating  in 
another  tangent,  B'F  ;  to  change  the  position  of  the 
Point  of  Tangent  B  of  the  first  curve,  so  that  the  second 
curve  may  terminate  m  a  given  parallel  tangent,  B"F'. 

Fig.  42. 

Let  X  be  the  required  new  position  of  B. 
"    0"  be  the  corresponding  position  of  0'. 
"    A'  =  A'O'B'  and  A"  =  A"0"B\ 

Since^the  radii  and  the  connecting  tangent  are  unchanged 
in  length,  and  all  rotate  together  about  0  as  a  centre,  0 "  will 
be  on  a  circle  passing  through  0',  described  with  a  radius 
00',  and  the  required  angle  BOX=  O'OO". 


SIMPLE   CURVES.  99 

In  the  diagram,  produce  0  'A'  and  draw  the  perpendicular 
OO,  and  let  a  =  the  angle  00' G.  Also,  draw  OK  parallel 
and  0"K  and  0 'H  perpendicular  to  B'O'.  In  the  triangle 
00' G  we  have 


cot  00 '£  =  —-,      or      cota=      ^~          (95) 
and 


cos  a 

The  angle  KOO1  =  00 'B'   =.«+  A'. 
The  angle  KOO"  =  00 "B"  ='a  -j-  A". 

^0  =  00".  cos  (a  -f  A"),      HO  =  00'.  cos  (a  -f  A'). 
.  \     1TJ5T  =  00'  [cos  (a  +  A")  -  cos  (a  -f  A')]  =  B  'F' 

B'F' 

cos  (a  +  A")  =  cos  (a  +  A')  +  -^7  (97) 

BOX=  O'OO"  =  (a  +  A')  -(a+  A")          (98) 

If  we  conceive  a  line  to  be  drawn  through  0  bisecting  the 
arc  O'O",  the  angle  it  makes  with  B"0"  is  a  mean  between 
B'0'0  and  B"0"0  ;  hence  the  chord  O'O",  perpendicular  to 
this  line,  makes  an  angle  with  O'P  perpendicular  to  B'O'  of 

PO'O"  =H(«+  A ')  +  (<*+  A")] 
and  since 

0'Pr=PO"  cotPO'O" 


5'^'coti[(a+  A ')  +  («+  A")]      (99) 

which  gives  the  distance,  measured  on  the  parallel  tangent, 
between  the  old  tangent  point  and  the  new. 

This  problem  occurs  in  practice  when  both  the  connecting 
tangent  and  the  radius  of  the  last  curve  are  at  their  minimum 
limit,  and  the  parallel  tangent  is  inside  of  the  old  one,  as  in 
the  figure.  Should  the  new  tangent  be  outside,  the  same  for- 
mulEe  apply,  only  changing  the  sign  of  B'F'  in  eq.  (97).  But 
in  this  last  case  it  is  usually  preferable  to  employ  problem 
§  136  or  §  137. 

Example. — A 1°  40'  curve  is  followed  by  a  tangent  of  200  ft., 
and  that  by  a  4°  curve  of  10  stations  ending  in  a  tangent ; 


100      .  FIELD   ENGINEERING. 

and  the  offset  to  the  given  parallel  tangent  is  80  ft.  on  the 
inside.     Kequired,  the  position  of  the  new  tangent  points  X 
and  B". 
Here  B  =  3437.87,  E'  =  1432.69,  BA  =  200,  B'F'  =  80. 

Eq.  (95)  R  +  R'  4870.56  log  3.687579 

BA'  200.  log  2. 301030 

.-.  a  2°  21'  log  cot  1.386549 

Eq.  (96)  a  2°  21'  log  cos  9.999635 

00'  :V:  3.687944 

Eq.  (97)  B'F      80  1.903090 

.01641  8.115146 

a+  A' 42°  21' cos  .73904 

a  -f-  A"  40°  56'  cos  .75545 

Eq.  (98)  BOX        1°  25'     .-.  BX=85  ft.  Ana. 

Eq.  (99)  PO'O"  41°  38'  30"  cot  1.12468  X  80  =  89.97  =  F'B  * 

156.  When  the  tangents  of  a  proposed  road  are  to  be  in 
general  much  longer  than  the  curves,  it  is  desirable  to  estab- 
lish the  tangents  first  in  making  the  location,  and  afterwards 
determine  suitable  curves.  On  the  other  hand,  if  the  curves 
necessarily  predominate,  they  should  be  first  selected  and 
adjusted  to  the-  ground  with  reference  to  grade  and  easy 
alignment,  and  afterwards  joined  by  tangents.  In  the  latter 
case  the  field  work  cannot  be  successfully  accomplished 
unless  the  location  has  been  previously  worked  out  upon  a 
correct  map  constructed  from  the  preliminary  surveys.  The 
map  sliould  show  contours  of  the  surface,  and  also  the  grade 
contour,  or  intersection  of  the  surface  and  plane  of  the  grade. 
In  side-hill  work  the  grade  contour  indicates  approximately 
the  degree  and  position  of  the  necessary  curves.  In  the  work 
of  selecting  proper  curves  upon  the  map,  templets  or 
pattern  curves  are  almost  indispensable.  The  templets  are 
cut  to  form  a  series  of  curves,  the  radii  being  taken  from 
Table  IV.  to  a  scale  corresponding  to  the  scale  of  the  map, 
which  ranges  from  400  to  100  feet  per  inch,  according  to  the 
difficulty  of  the  location.  The  templets  should  represent 
convenient  curves,  or  those  in  which  the  number  of  minutes 


SIMPLE   CURVES. 


KM 


per  station  bear  a  simple  ratio  to  100.  Curves  of  50'  and 
multiples  of  50'  are  most  convenient;  40'  curves  and  multi- 
ples standing  next  in  order,  and  30'  curves  and  multiples 
next. 

TABLE  OF  CONVENIENT  CURVES. 


D. 

Ratio  of  Min. 
to  Feet. 

D. 

Ratio  of  Min. 
to  Feet. 

D. 

Ratio  of  Min. 
to  Feet. 

50' 

1    2 

40' 

2:5 

30' 

3:10 

1°  40' 

1    1 

1°  20' 

4:5 

1°  00' 

3:5 

2°  30' 

3    2 

2°  00' 

6:5 

1°  30' 

9:10 

3°  20' 

2    1 

2°  40' 

8:5 

2°  00' 

6:5 

4°   10' 

5    2 

3°  20' 

2:1 

2°  30' 

3:2 

5°  00' 

3    1 

4°  00' 

12:5 

3°  00' 

9:5 

5°  50' 

7    2 

40  40/ 

14:5 

3°  30' 

21  :  10 

G°  40' 

4    1 

5°  20' 

16:5 

4°  00' 

12  :  5 

7°  30' 

9    2 

6°  00' 

18:5 

4°  30' 

27:  10 

8°  20' 

5    1 

6°  40' 

4:  1 

5°  00' 

3:  1 

9°   10' 

11    2 

7°  207 

22:5 

5°  30' 

33:10 

10°  00' 

6    1 

8°  00' 

24:5 

6°  00' 

18:5 

After  drawing  the  curves  and  tangents  upon  the  map,  the 
tangent  points  and  central  angles  are  carefully  determined, 
the  latter  being  compared  with  the  lengths  of  the  curves  ob- 
tained by  a  pair  of  stepping  dividers  set  precisely  by  scale  to 
the  length  of  one  station.  Field  notes  are  then  prepared  from 
the  map,. and  if  the  work  has  been  well  done  these  notes  may 
be  followed  in  the  field  with  scarcely  any  alterations. 

No  ordinary  protractor  will  measure  the  angles  closely 
enough  for  this  purpose  ;  it  is  better  to  use  a  radius  as  large 
as  convenient,  of  50  parts.  The  chord  of  any  arc  drawn  with 
this  radius  equals  100  times  the  sine  of  one  half  the  angle 
subtended. 

The  importance  of  having  absolutely  straight-edged  rulers 
in  such  work  is  obvious.  In  case  a  very  long  line  is  to  be 
projected  upon  the  map,  it  is  well  to  use  a  pieqe  of  fine 
sewing  silk  for  the  purpose.  See  §§  53,  54. 


FIELD   ENGINEERING. 


CHAPTER  VI. 
COMPOUND  CURVES. 

••'••-      A.  Theory. 

157.  A  compound  curve  consists  of  two  or  more  consecu- 
tive circular  arcs  of  different  radii,  having  their  centres  on 
the  same  side  of  the  curve  ;   but  any  two  consecutive  arcs 
must  have  a  common  tangent  at  their  meeting  point,  or  their 
radii  at  this  point  must  coincide  in  direction.     The  meeting 
point  is  called  the  point  of  compound  curve,    or    P.C.C. 
Compound  curves  are  employed  to  bring  the  line  of  the  road 
upon  more  favorable  ground  than  could  be  done  by  the  use 
of  any  simple  curve. 

When  a  compound  curve  of  two  arcs  connects  two  tangent 
lines,  the  tangent  points  are  at  unequal  distances  from  the 
intersection  or  vertex,  the  shorter  distance  being  on  the  line 
which  is  tangent  to  the  arc  of  shorter  radius. 

158.  Let  VA,  VJ3  (Fig.  43)  be  any  two  right  lines  inter- 
secting at  V,  and  let  A  be  the  deflection  angle  between  them. 
Let  A  and  B  be  the  tangent  points  of  a  compound  curve  ( VA 
less  than  VB),  and  let  AP,  PB  be  the  two  arcs  of  the  curve. 
The  centre  Oi  of  the  arc  AP  will  be  found  on  AS,  drawn  per- 
pendicular to  VA  ;  the  centre  02  of  the  arc  PB  will  be  found 
on  BS  produced  perpendicular  to  VB ;  and  the  angle  ASB 
will  evidently  equal  A.     Join  VS,  and  on  VS  as  a  diameter 
describe  a  circle;  it  will  pass  through  the  points  A  and  B, 
since  the  angles  VAS,  VBS  are  right  angles  in  a  semicircle. 
Draw  the  chord  VQ,  bisecting  the  angle  AVB,  and  join  AQ, 
BQ.     Then  AQ,  BQ  are  equal,  since  they  are  chords  subtend- 
ing the  equal  angles  AVQ,  BVQ.     From  Q  as  a  centre,  and 
with  radius   QA,  describe  a  circle  ;    it  will  cut  the  tangent 
lines  at  A  and  B,  and  also  at  two  other  points  G  and  Y,  such 
that  VQ  =  VA,  and  VY=  VB.     Hence  BG  =  AY,  and  the 
parallel  chords  AO,  BY  are  perpendicular  to  VQ.     Join  AB; 
then  AQB  =  ASB  —  A  ,  since  both  angles  are  subtended  by 
the  same  chord  AB. 

In  the  triangle  VAB,  the  sum  of  the  angles  at  A  and  B  is 
equal  to  the  exterior  angle  A  between  the  tangents  ;  while 
their  difference  (A  —  B)  is  equal  to  the  angle  at  the  centre  Q 


COMPOUND   CURVES. 


103 


subtended  by  the  chord  BG,  which  is  the  difference  of  the 
sides  (VB  -  VA).  For  the  angle  VAB  =  VAG  +  GAB.  and 
the  angle  VBA  =  VBT  -  ABY.  But  VAG  =  VBY  and 
GAB  =  ABY,  and  by  subtraction  VAB  -  VBA  =  2GAB  = 
GQB,  since  A  is*  on  the  circumference  and  Q  at  the  centre. 

159.  THEOREM. — The  circle  YAGB,  whose  centre  is  Q,  ie 
the  locus  of  the  point  of  compound  curve  P,  whatever  be  the 
relative  lengths  of  the  arcs  AP,  PB  composing  the  curve. 


FIG.  43. 

On  the  circle  YAGB,  and  between  A  and  G,  take  any  point 
P,  and  on  vl/Sfind  a  centre  0M  from  which  a  circular  arc  may 
be  drawn  cutting  the  circle  at  A  and  P  ;  also  on  B8  produced 
find  a  centre  02,  from  which  a  circular  arc  may  be  drawn 
cutting  the  circle  at  B  and  P.  Join  PQ,  POi  and  POS. 
Since  when  two  circles  intersect,  the  angles  are  equal  be- 
tween radii  drawn  to  the  points  of  intersection,  QPOi=  QAOi 


104  FIELD   ENGIHEEKIKG. 

\  • 

and  QP02  =  QBO*.  Draw  the  chord  QS  and  it  subtends  the 
equal  angles  QAOl  =  QBO*.  Hence  QP01  =  QPO*  and  the 
radius  P0l  coincides  in  direction  with  the  radius  P02,  which 
is  the  condition  essential  to  a  compound  curve. 

Now,  if  we  imagine  another  point  P '  to  be  taken  on  QP  or 
on  QP  produced,  and  the  arcs  AP'  BP',  drawn  from  centres 
found  on  A8  and  B8,  it  is  evident  that  the  equality  of  angles 
found  in  respect  to  P  could  not  exist  in  respect  to  P.  Hence 
the  arcs  would  intersect  in  P'  at  some  angle  0iP02  and  would 
not  form  a  compound  curve.  Therefore,  Q.  E.  D. 

16O.  THEOREM. — In  any  compound  curve  the  radial  lines 
passing  through  the  three  tangent  points  A,  P,  and  B  are  all 
tangent  to  a  circle  having  the  point  Q  for  its  centre,  and  for  its 
diameter  the  difference  of  the  sides  VB  and  VA. 

Draw  the  three  lines  QM,  QN,  QL  perpendicular  to  the 
radial  lines  BO^,  AS,  and  P02  respectively.  Then  the  three 
right-angled  triangles  BQN,  PQL,  and  AQM^are  equal,  since 
BQ  =  PQ  —  AQ  =  radius  of  the  circle  AGB,  and  the  angles 
at  B,  P,  and  A  are  equal  by  the  last  theorem.  Hence  QM  = 
QL  =  QN,  and  if  a  circle  be  described  with  this  radius  about 
Q,  the  three  lines  BO^,  P02,  and  AOi  produced  will  be  tan- 
gent to  it.  Draw  Ql  perpendicular  to  VB;  it  will  bisect  the 
chord  GB  in  7;  and  QN  =  BI  —  $BG.  Hence  the  diameter 
2QN=  BG  =  VB  —  VA;  which  was  to  be  proved. 

Corollary  1.  The  compound  curve  intersects  the  circle  AGB 
in  the  point  P,  at  an  angle  equal  to  half  the  difference  of  the 
angles  VAB,  VBA.  For  QPL  =  QBN=  BQI  =  iBQG.  The 
arc  AP  is  exterior,  and  the  arc  PB  interior  to  the  circle 
AGB. 

Cor.  2.  Since  both  centres  are  on  the  line  PL,  the  position 
of  the  point  P  fixes  the  lengths  of  the  radii  of  a  compound 
curve.  As  P  is  moved  toward  G  both  radii  are  increased, 
until  when  P  reaches  G-,  AOi  becomes  AK,  a  maximum,  while 
B0<i  becomes  infinite.  As  P  moves  toward  A  both  radii  are 
diminished,  but  the  least  value  of  the  arc  AP  depends  upon 
the  least  radius  allowed  on  the*  road.  If  in  the  diagram  we 
make  AOi  equal  to  the  least  radius  allowed,  a  right  line  drawn 
through  the  point  0^  tangent  to  the  circle  LMN  fixes  the 
corresponding  minimum  value  of  the  arc  AP,  and  also  of 
the  radius  BO?  for  given  values  of  VA,  VB,  and  A.  Be- 


COMPOUND   CURVES.  105 

tween  these  limits  any  desired  values  of  the  radii  may  be  em- 
ployed. 

Cor.  3.  In  the  triangle  SOi  02,  the  sum  of  the  two  central 
angles  AOiPand  P02B  is  equal  to  the  exterior  angle  ASB  = 
A ;  consequently,  as  the  central  angle  of  one  arc  is  increased 
by  any  change  in  the  position  of  the  point  P,  the  central 
angle  of  the  other  will  be  diminished  an  equal  amount. 

Cor.  4.  Only  one  value  of  the  angle  AOiP  is  consistent  with 
a  given  value  of  the  radius  A0lt  since  both  depend  on  the 
variable  position  of  the  line  PL;  and  for  the  same  reason  only 
one  value  of  the  angle  BOiP  is  consistent  with  a  given  v^lue 
of  the  radius  BO?.  Hence  only  one  radius  or  one  central 
angle  can  be  assumed  at  pleasure,  the  remaining  parts  being 
deducible  therefrom  in  terms  of  the  sides  VA,  VB,  and  the 
angle  A. 

B.  General  Equations. 

161.  Let  81  =  the  side  VA,  83  =  the  side  VB 

Let  J?i  =  the  radius  AOi  Jf?a  =  the  radius  B0<t 

"    y  =  diff.  VAB  -  VSA,  A  =  the  sum  VAB+  VBA 

"  AI  —  central  angle  AOiP,  A2  —  central  angle  BO^P. 

In  the  triangle  BQI,  cot  BQI  =  -~.     But   10  =  VI  X 

-£>_/ 

cot  /  Q  V  =  i(89  +  /Si)  cot  I  A  ,  and  BI  =  K&  -  Si). 

cot  \y  =  -g~^  cot  i  A  (100) 

By  Cor.  3,  Ai  +  A2  =  A  (101) 

In  the  triangle  AQM,  AOi  =  AM  —  MOi.  But  AM  = 
MQ  cot  \y,  and  MOi  =  MQ  cot  £AI. 

Si  =  i(Si  —  Si)  (cot  \y  —  cot  |  A  i)  > 

(102) 
Similarly,      jRa  =  4(5a  -  Si)  (cot  \y  +  cot  |  A  2) ) 

Subtracting, 

J2a  -  JRi  =  K/S'a  -  Si)  (cot  i  A  2  -f  COt  i  A  0        (103) 


106  FIELD   E 


cot  i  A  i  =  cot  \y  - 
From  (102),  \  (104) 

In  the  triangle  ABG, 

^^      AB  sin  BAG 

sinAGV 
or 


by  which  we  find  ^(S*  —  &),  when,  instead  of  the  sides  and 
A  ,  we  have  given  AB,  and  the  angles  VAB  and  VBA. 


From  (103), 


-|  A 


T-> 

•     ox  +  COt  ^  A 
—  ft) 


! 

From  (102),  }•  (107) 


From  (100),     J<A  +  ft)  =          =  (108) 


^2  and  &  are  found  by  adding  and  subtracting  the  values 
found  by  eqs.  (106),  (108). 

From  (105),         t^  =  ««L^lM    •'         ,     (109)     - 

which  may  be  used  instead  of  (108)  when  the  sides  are  not  re- 
quired.    VAB  =  i-(  A  +  r)  and  VBA  =  |(  A  -  r). 

162.  Given  :  the  sides  VA  =  Si  and  VB  =  82  and  the 
angle  A;  assuming  the  shorter  radius  jRlt  to  find  Ai,  A  a, 
and  jR2. 

Use  equations  (100),  (104),  (101),  (102),  and  (18). 

Example.—  Let  VA  -  1899.90,  VB  =  1091.12,  A  =  74°,  and 
assume  -Bi  =  955.37. 


COMPOUND   CURVES.  107 

(100)  K&  4-  S>  )  1495.  51  log  3.  174789 

£(&-&)  404.39  "  .2.606800 

"  0.567989 
A         37°  cot  •"  0.122886 


.-.  \y          11°  31'  01".5  cot  4.90769  "  "  0.690875 

(104)  R,  (D  =  6°)  "  2.980170 

&-&)  "  2.606800 


2.36249    "  0.373370 


.-.     A  A!  21°  27'  cot  2.54520 

(101)  i  A  37° 

A2  15°  33'  "    3.59370 

y  "    4.90769 


jt-to  | 
(102)  \ 


8.50139    "  0.929490 
"  2.606800 


.  •.     R*  (D  =  1°  40')  3.536290 

(18)  .'.  Ax  =  42°  54',  Zi  =  715;  Ae  =  31°  06',  Z8  =  1866. 

163.  Given:   the  line  AB,  and  tJie  angles  VAB,  VBA; 
assuming  tJie  longer  radius  R^,  to  lind  A2,  Aj,  and  Ri. 

Example.— Let  AB  =  2437.82,  VAB  =  48°  31',  VBA  =  25°  29', 
and  assume  J?2  =  3437.87. 

(105)  \AE  1218.91  log  3.085972 

\Y  11°  31'  sin  "     9.300276 


"    2.886248 
37°  '"    "    9.779463 


"    2.606785 
(104)  &  3.536289 


8.50166     "    0.929504 
11°  31'  cot  4.90785 


.-.     iAt  15°  33*  cot  3. 59381 

(101)  i  A  37° 


.*.     iAt  21°27f  cot  2,54516 

(102)  \y  "  4.90785 


2.36269  log  0.373407 
2.606785 


(D  =  66)  2.980192 


108  FIELD 

164.  Usually  a  compound  curve  is  fitted  by  trial  to  the 
shape  of  the  ground,  after  which  it  may  be  desirable  to 
calculate  the  sides  VA,  VB,  or  the  line  AB,  and  the  angles 
VAB,  VBA. 

Example.— From  the  point  of  curve  A,  a  6°  curve  is  run 
715  feet  to  the  P.  C.C.;  thence  a  1°  40'  curve  is  run  1866  feet 
to  the  P.T.  Required,  the  sides  VA,  VB,  and  the  line  AB, 
and  angles  VAB,  VBA.  Here  fit  =  935.37,  AI  =  42°  54', 
J2a  =  3437.87,  A2  =  31°  06'. 


(106)  #,-!?,     2482.50  log  3.394889 

iA,  21°  27'        cot  2. 5451 6 

*  A,  15°  33'  "  3.59370 

6.13886        "    0.788088 


-&)  404.39  "    2.606801 

"    2.980170 


2.36248 

0.373369 

*A, 

21°  27'        cot  2,  54516 

•%    ir 

ll°31'0r.7  "4.90764 

0.690873 

(108)  £(&  -  AS 

2.606801 

« 

3.297674 

|A 

37°                                cot  " 

0.122886 

...     ^-[-/S 

0  1495.51 

3.174788 

$a 

1899.90 

A 

1091.12 

FA# 

48°  31' 

F#A 

25°  29' 

(109)  |(&  -  £ 

)                                                            " 

2.606801 

IA 

37°                                 sin  " 

9.779463 

2.386264 

Ir 

11°  31'  Ol'.T                  sin  " 

9.300294 

.-.     |^4^ 

1218.91 

3.085970 

^li? 

2437.82 

165.  Given  :  the  radii  J?,,  ^2,  ^  «w//?e  A, 
K4,  or  VB,  to  find  ffo  other  side  and  the  central  angles 
A  a."   Fig.  43, 


COMPOUND   CURVES.  109 

In  the  triangle  AMQ,  A0t  =  AM  -  M0l  =  IQ  -  MQ  cot 
MOiQ;  or 

J21  =  $(S<>  +  S,)  cot  4  A  -  i(&  — &)  cot  iAi 
whence 

i(&  +  £,)  =  -K&  — £,)~cot  i  A,  tan  4 A  +  -Bi  tan  |A 
By  eq.  (106) 


Substituting  this  above,  subtracting  and  reducing 

Si  =  (l?a  —  -Ri)  sin  i  Aa  —  -  -+-  Ri  tan  iA 

But£(A  —  A])  =-jAa  and  2 sin2  £Aa  =  vers  A2,  whence 


c  _  (1?.  -  l?i)  vers    Aa  +  IZi  vers  A  1() 

1  sin  A 


Transposing, 


#,  sin  A  —  12i  vers  A 

vers  A ,  =—      — j5 s (HI) 

ft*  —Mi 


Similarly,  from  the  triangle  BQO* 

Ri  =  i(5a  +  -S,)  cot  i  A  -f-  -K^a  —  ^i)  cot  i 
from  which  and  eq.  (106)  we  derive 

_  R*  vers  A  —  QRa  —  #0  vers  At 
sin  A 

and 

Ri  vers  A  —  &  sin  A 
vers  AI  =  — — — ^c- n ' 


110  FIELD 


Example.—  Given  :  VA  =  Si  =  1091.12,   A  —74°,  and  the 
radii  Rt  =  955.37,  R*  =  3437.87,  to  find  AI,  A2,  and  £a. 


(Ill)  8,            1091.12  log  3.037873 

A                               74°  sin  "     9.982842 

1048.85  "    3.020715 

Si  "    2.980170 

A                              74°  vers  "    9.859956 


692.03  "    2.840126 


356.82  "    2.552449 

"    3.394889 


A  2  31°  06'  vers  "     9.157560 


A!  42°  54' 


....  " 

9.427254 

3.394889 

663.96 
vers  " 

2.822143 
3.536289 
9.859956 

2490.26 

3.396245 

3.261572 
9.982842 

1826.30 
sin  " 

A 
.-.     8,  1899.90  "    3.278730 

166.  Given  :  one  side,  and  the  radius  and  central  angle  of 
the  adjacent  arc,  to  find,  the  other  radius  and  side. 
From  eqs.  (Ill),  (113)  we  have 

81  sin  A  —  Ri  vers  A 


vers  A2 

(114) 
R?  vers  A  —  81  sin  A 


by  one  of  which  the  required  radius  may  be  found  ;  the  required 

side  is  then  found  by  eq.  (110)  or  (112),  as  in  the  last  problem. 

Example.—  Given  :  VA=8i  =  1091.12  A  =  74°,  RI  =  955.37 

and  A  ,  =  42°54'  ;  to  find  R*         Aa  =  74°  -  43°  54'  =  bl°  06'. 


COMPOUND   CURVES. 


Ill 


(114)  & 


A2 


1091.12 


955.37 


2482.52 
3437.89 


74° 


74° 


31°06 


log    3.037873 
"  sin  9.982842 


1048.85 


692.03 
356.82 


3.020715 


2.980170 
vers  9.859956 


2.840126 


2.552449 
vers  9. 157556 


3.394893 


FIG.  44. 


Otherwise  :  Fig.  44.  If  convenient  in  the  field,  a  tan- 
gent PF3  may  be  run  from  the  point  P  to  intersect  the 
farther  tangent.  The  distance  PF3  multiplied  by  cot  iA2 
will  equal  the  radius  JKa  by  eq.  (25). 

167.  Remarks.— It  the  first  arc  AP  be  produced  to  G, 
Fig.  44,  so  that  AO^G  =  A,  then  G  is  the  tangent  point  of  a 
tangent  parallel  to  VB,  and  by  §137,  the  tangent  point  B  must 
be  on  the  line  P&  produced.  Conversely,  if  the  point  B  is 
assumed,  and  the  arc  AG  given,  the  point  P  must  be  on 
the  line  BG  produced.  The  radius  R*  may  be  found  by 


112  FIELD 

737) 

.R3  =  — ; — ,  BP  being  measured  on  the  ground  ;  or  by 

&  sin  -£A2 

similar  triangles  J?2  :  72i  ::  BP  :  GP. 

The  distance    YD,  Fig.  43,  from  the  vertex  to  the  circle 
AGB  is  expressed  by  the  formula 


If  the  point  P  falls  at  D,  then  YD  is  also  the  distance  of  the 
curve  from  the  vertex  measured  on  the  line  VQ.  But  when 
P  falls  at  D,  the  radius  P02  is  perpendicular  to  the  line  AB, 
and  AI  =  VAB,  and  A2  =  VBA.  When  AI  is  greater  than 
VAB,  the  arc  AP,  being  exterior  to  the  circle,  cuts  the  line 
YD;  but  when  At  is  less  than  VAB,  the  arc  PB  cuts  the  line 
DQ. 

If  the  line  02P  produced  passes  through  V,  we  have 

sin  Q  VL  =  8>-~      sin  i  A  (116) 


giving  AI  =  iA  +  QVL  and  A2  —  4-A  — 

When   AI  is  greater  than  this,  we  have  for  the   external 
distance  of  the  vertex 

El  =  &  ex  sec  AOiV 
in  which  the  angle  AO^.  Fis  found  by  the  formula  tan  AOtY= 

73 

—  ,  and  Ei   is  measured  on  a  line  V0lt  making  the  angle 

Si 

AVO,  =  90°  -  AO^V. 

When  A  i  is  less  than,  (-J-  A  -j-  Q  VL),  we  have  similar  expres- 
sions with  respect  to  the  arc  BP  and  centre  Oa. 


168.  To  locate  a  compound  curve  when  tJie  point  of  com- 
pound curve  it  inaccessible.  Fig.  45. 

Each  arc  being  in  itself  a  simple  curve  is  located  as  such. 
When  the  P.C.C.  is  accessible,  the  transit  is  placed  over  it, 
and  the  direction  of  the  common  tangent  found,  from  which 
the  second  arc  is  then  located. 

When  the  P.C.C.  is  not  accessible,  the"  common  tangent 
V\  Fa  may  be  found  by  locating  the  points  Y\  and  Fa,  which 
may  be  easily  done,  since  Y\A  —  Y\P  —  R\  tan  £AI,  and 


COMPOUND   CURVES. 


113 


V'tP  ==  It*  tan  1  A  2,  from  which  each  arc  may  then 
be  located  by  offsets  or  otherwise,  as  in  the  case  of  simple 
curves. 

Should  the  points  V\  F2  be  obstructed,  the  common  tangent 
may  be  found  by  an  offset  IIG  =  LP  from  any  convenient 
point  11,  for  knowing  the  angle  HO^P,  we  have  HG  =  J2i 
vers  HOiP,  and  GP  =  A',  sm  HO,  P. 

If  the  entire  tangent  Pi  F2  is  too  much  obstructed  for  use, 
the  parallel  line  HK  may  be  employed,  observing  that  the 

LP 
angle  POtK  is  found  by  vers  P02K  =  -=-,  and  the  distance 

LK  by  LK  =  R»  sin  PO^K,  by  which  a  point  K  on  the  second 
arc  is  found  having  a  tangent  offset  KI  =  HG. 


FIG.  45. 


FIG.  46. 


Should  the  line  HK  be  also  obstructed,  we  may  run  the  in- 
verted curve  HP'  —  HP  and  P'K  =  PKto  find  the  point  K 
from  which  so  much  of  the  second  arc  as  is  accessible  may  be 
located. 

C.  Special  Problems  in  Compound  Curses. 

169.  Given:  a  compound  curve  ending  in  a  tangent;  to 
change  the  P.C.C.  so   that  the  curve  may  end  in  a  given 
parallel  tangent.    Fig.  46. 
Let  APE  be  the  given  curve  ending  in  VB, 
"    VI?  be  the  given  parallel  tangent, 
"  p  =  perpendicular  distance  between  tangents. 
It  is  required  to  change  the  point  P,  and  with  it  the  values 
of  AI  and  A  2,  so  that  with  the  same  radii  KI  and  J22  the  new 
curve  APB  may  end  in  the  parallel  tangent  VB. 


114  FIELD 

a.   WJien  the  tangent  VB'  is  inside  of  VB  : 

and  in  the  diagram  draw  0j(?  perpendicular  to  BO*-,  then 
GO*  =  0i02  cos  A«,  KO-i  =  0i02  cos  A 2'.  Subtracting, 
since  0;02  =  0i02'  =  (^  -  -#1),  and  KOi  -  GO*  =  GB  - 
KB'  =  p> 

p  =  (Ri  —  Rt)  (cos  A  ar  —  cos  A  2) 
whence 

COS  A  2'  =   7>          „    -4-  cos  A  a  (117) 

Hz  —  K, 

POiP'  =  ( A2  —  A  2')  and  the  point  P  is  advanced. 
h.  FAm  tAe  teu»src»<  F'P'  w  outside  of  VB: 

p  =  (J?a  —  Rt)  (COS   A  2  —  COS  A3') 

•whence 

COS  A  a'  =  COS  A  2  —  (118) 

P0jP'  =  (A 2'  —  A 2)  and  the  point  P  is  moved  back  and  the 
arc  AP  diminished. 


FIG.  47. 

7^  case  the  curve  terminates  with  the  arc  of  shorter 
radius,  or  Rl  follows  R*.    Fig.  47. 
c.  When  VB  is  inside  of  VB: 

p  =  (Rz  —  R^)  (ccs  AI  —  cos  AO 
whence 

cos  Ai'  =  cos  AI  — =: =r- 


PO*P'  —  (AI'  —  Ai)  and  the  point  P  is  moved  back. 


COMPOUND   CURVES.  115 

d.   When  V'B'  is  outside  of  VB: 

p  =  (Mi  —  RJ  (cos  A  /  —  cos  A  i) 
whence 

cos  A,'  =cos  Ai  -f  -^ sr 


P03P'  —  (Ai  —  Ai')  and  the  point  P  is  advanced. 

Example.— Let  R  =  2292.01,  Ri  =  1432.69,  A2  =  28°,  and 
p  =  20.07  inside  of  VB  ;  case  a. 

p  20.07  log  1.302547 

(117)  #2  -  Ri    859.32  "    2.934155 

.023356        "    8.368392 
A  3  '  28°  cos  .88295 

.'.      A'a  2JT  "  7906306 

.-.     PO.P'  3° 

17O.  Given:  a  compound  curve  terminating  in  a  tangent, 
to  change  the  P.C.C.  and  also  the  last  radius,  so  that  the 
curve  shall  end  in  a  parallel  tangent  at  a  point  on  the 
same  radial  line  as  before.  Fig.  48. 


FIG.  48. 

Let  APB  be  the  given  curve  ending  in  the  tangent  VB\  let 
V'B'  be  the  given  parallel  tangent;  and  let  p  —  BE'  =  ///= 
the  perpendicular  distance  between  tangents. 

It  is  required  to  change  the  point  P  to  P',  and  also  the 
value  of  R*  to  RJ,  so  that  the  new  curve  may  end  in  V'B'  at 
B'  inside  of  VB  on  the  same  radial  line  #02. 

In  the  diagram  produce  the  arc  AP  to  Q  to  meet  Oi  G 
drawn  parallel  to  025;  then  POiG  =  A2.  Draw  the  chord 
PB,  and  it  will  pass  through  G.  Lay  off  the  distance  p  from 


116  FIELD   ENGINEERING. 

B  on  BO*  to  find  B1  ;  draw  B'G  and  produce  it  to  intersect 
the  arc  APG  in  P'.  Then  P'  is  the  P,C.C.  required.  Join 
P  '  Oi  and  produce  it  to  meet  BO-2  produced  in  0?.  Then 
P'0a'  =  l?'0a'  =  R*  the  new  radius,  with  which  describe  the 
arc  P'B'. 

By  Geom.  Tab.  I.  18  : 

PBV  =  i  P03Jff  =  iAa,  and  #£'  F  =  ±P'0*B  =  |A2'. 

PGP'=BGB'  =  -K  A2  -  A2') 

Draw  OiJST  perpendicular  to  J502. 

Then  0^=  B'ff=BI=  Oi02sin  A2  =  (5a  -  -Bi)  sin  A2 

GI-  P 


.:,,  .   tan  iA2'  =  tan  |A2  -  ^-__I---  (121) 

In  the  triangle  Oj0202' 

sin  A  3'  :  sin  A2  ::  0^  :  OM  ::  (R*  -  E,)  :  (H*  -  & 

B9'-Sl=*±J±.(Ba-Sl) 

sm  A2 

^2'-(^2-JR1)^|27  +  ^  (122) 

Sin   Aa 

IfB'  V  were  outside  of  VB; 


W hen  the  smaller  radius  Ri  follows  Rz :   If  the  given 
tangent  B'V  is  inside  of  BV.    Fig.  49. 

•    tan  iA/  =  tan  IA,  +  T        -^--5-—         (124) 


COMPOUND   CURVES. 


11' 


IfB'V  is  outside  ofBV: 

tan  $Aif  =  tan  |AI  — 


—  Ri)  Bin 

sin  Ai 

sin  AI' 


(126) 
(135) 


FIG.  49. 


.—m.  48. 


Let  R*  =  2292.01        p  =  20.07  inside. 
"  ft  =  1432.69     Aa  =  28° 


(121) 

(122) 
( 

( 

ft  —  Hi 

A2 

p 

tan  iAa 

tan  |Aa' 
AV 

R2  -  ft) 
ft'  -  ft) 

20.07 

.04975 
.24933 

log  2.934155 
28°                 log  sin  9.671609 

2.605764 
1.302547 

8.696783 

-    \ 

11°  17' 
22°  34'                sin  9.584058 
2.934155 

.19958 

1051.25 
1432.69 

3.350097 
28°                      sin  9.671609 

3.021706 

Ans.        Rz'        2483.94  .  •.  D  =  2*  18'  25" 

PO,P  =  28°-  22°  34'  =  5°. 26'  .'.  PP'  =  135.83  ft. 


FIELD 


Example  2.—  Fig.  49. 


Let  R*  =  2292.01        p  =  20.07  inside. 


(124)  JB,  - 


tan 


^  =  1432,69 
859.32 


20.07 


.03247 
.42447 

.45694 


Ai 


JZ, 


46C 

46° 

23° 

24°  33^ 
49°  07' 

46° 


log  2.934155 
log  sin  9. 856934 

2.791089 
1.302547 

8.511458 


log  sin  9.878547 
2.934155 

3.055608 
log  sin  9.856934 


817.60 
2292.01 


Ans. 


S  =  1474.41  .  •.  D  =  3°  53'  12" 


P02P'  =  A  /  -  A  :  =  3°  07'  .  '  .  arc  PP'  =     il1^  =  124.67  ft. 

Observe  that  in  either  figure  both  tangents  must  be  on  the 
same  side  of  the  point  G,  in  order  to  a  solution. 


FIG.  50. 

171.  Given:  a  compound  curve  ending  in  a  tangent,  to 
change  the  last  radius  and  also  the  position  of  the  P.C.C., 
so  that  the  curve  may  end  in  the  same  tangent.  Fig.  50. 


COMPOUND   CURVES.  119 

I.   When  the  curve  ends  with  the  greater  radius  7?2. 

Let  APB  be  the  compound  curve  in  which  MI  MI  A  i  and 
A  a  are  known. 

In  the  diagram  draw  the  chord  PB  and  produce  the  first 
arc  AP  to  meet  it  in  6r;  draw  OiG,  and  produce  it  to  meet  the 
tangent  in  K.  Then  by  §  137  OiK  is  parallel  to  0ZB,  and  by 
eq.  (57) 

OK  =  (J2»  -  #0  vers  A2  (127) 

If  we  assume  P'  as  the  new  P.  C,  G.,  we  have  A  a'=  P'O-t'B', 
and  the  chord  P'G  produced  will  intersect  the  tangent  at  the 
new  point  of  tangent  B\  and  J?03'  =--Ra'.  Similar  to  eq.  (127) 
we  have 

OK—  (Ri  —  Hi)  vers  A2' 

and  equating  the  two  expressions,  we  obtain 


vers  A  a  vers  Aa 

If  we  assume  RaJ  we  have 

vers  A2'  =  -|^~-  vers  A2  =  =£*  (129) 

ft*  —  Mi  Mi  —  Mi 

In  the  two  right-angled  triangles  BKG  and  B'EG,  we  have 

BK= 
B'K  = 

and  by  subtraction, 

BB'  =  GK  (cot  \  A  a'  -  cot  I  A  3)  (130) 

in  which  GK  is  obtained  from  eq.  (127). 

When  BB'  as  given  by  eq.  (130)  is  negative,  the  p'oint  B'  falls 
between  5  and  F. 

#"  we  assume  ;fc  distance  BB'  <w  £&6  tangent,  we  have 
from  the  last  equation, 

T>  7> 

COt|A2'=COtiA2   ±        ~  (131) 


FIELD 

G^being  obtained  from  eq.  (127)  and  K  from  e*  (US) .    ^ 
eq.  (131)  use  the  +  sign  when  B'  is  beyond  B  as  in  the  Fig.^5( 
II.   When  the  given  curve  ends  with  the  smaller  radius 
B,.    Fig.  51. 

/V 


FIG.  51. 


We  have  by  a  similar  reasoning 

GK  -  (B*  -  Bi)  vers  A 


(133) 
vers 


(134) 


=  GK  (cot  1  A  ,  -  cot  \  A  i') 

=  COt^Ai    ± 


- 


using  the  -  sign  when  B'  is  beyond  #. 

Example.—  Fig-  51. 

Let  ^2  -  2292.01,    A  -  1482.69,    A,  =  46°,  and  le  ;  the 
P  C  C  be  moved  back  200  feet  from  P  to  P  ;  hence  P02  P 
5°'  and  A/  =  51°;  to  find  the  new  radius  B,'  and  the  d 
BB'. 


COMPOUND  CURVES. 


Eq.  (132)  ft*  - 

Ai 

.-.  GK 
eq.  (133)  AI' 

R*  -  Ei' 
K* 

.'.  Ei' 
eq.  (135)  GK 

COt  -£  A  i 
COtiAi' 

.-.  BH 

Ei    859.32 

707.85 
2292.01 

46° 
51° 

and  D  =  3° 

23° 

25°  30' 

log          2.934155 
"    vers  9.484786 

log          2.418941 
"    vers  9.  568999 

2.849942 

37' 
log          2.418941 

log          9.413819 
1.832760 

1584.16 

2.35585 
2.09654 

0.25931 

• 

68.04 

172.  Given:  a  compound  curve  ending  in  a  tangent,  the  last 
radius  being  the  greater,  to  change  the  last  radius  and 
also  the  position  of  the  P.C.C.  so  that  the  curve  may  end  at  the 
same  tangent  point,  but  with  a  given  difference  in  the 
direction  of  the  tangent.  Fig.  52. 


Fia.  52. 


Let  APS  be  the  given  compound  curve,  POi  —  Ei  and 
P02  =  £2  >  Ei. 

LetF'-Sbe  the  new  tangent,  and  the  angle  V~BV  =  *,  the 
given  difference  in  direction  :  to  find  BO*  =  E*',  BO*P'  — 
A  a'  and  the  angle  POiP\ 


122  FIELD    ENGINEERING. 

We  have 

BO,  -  0,0,  =  It,  -  CR2  -  R,}  =  R! 
BO*  -  0i02'  =  R2'  -  (.Ra'  —  Ri)  =  Rt 

From  which  we  see  that  whatever  may  be  the  value  of  the 
new  radius,  the  difference  of  the  distances  from  B  and  0i  to 
the  new  centre  is  constant,  and  equal  to  Ri.  We  therefore 
conclude  that  the  centres  02  and  0,'  are  on  an  hyperbola  of 
which  B  and  Oi  are  the  foci,  and  Ri  the  major  axis. 

This  suggests  an  easy  graphical  method  of  solving  the 
problem. 

Through  B  draw  a  line  perpendicular  to  the  new  tangent 
V'B  which  will  give  the  direction  of  the  required  centre  02'. 
On  this  line  lay  off  BK  equal  to  Ri,  and  since  (#a'  —  -Bi)  = 
0i  0a'  =  KO*,  if  we  join  KOi,  the  triangle  K0-2'0l  is  isosceles; 
therefore  bisect  K0t  and  erect  a  perpendicular  from  the  mid- 
dle point  to  intersect  the  line  BK  produced  in  0a'.  Draw  02'0, 
and  produce  it  to  intersect  the  arc  AP  (produced  if  necessary) 
in  P'.  Then  P'  is  the  new  P.O.C.  required,  and  2?0a'  = 
P'Oa  =  Ri.',  the  new  radius. 

The  analytical  solution  is  as  follows  : 
Adopting  the  usual  notation  of  the  hyperbola 

0 
Let  2a  =  Ri     =  the  major  axis, 

"  2c  =  BO  ,  =  the  distance  between  foci. 

Produce  the  arc  AP  and  through  B  draw  the  tangent  BH, 
and  join  HOi  =  Ri.  Then  in  the  right-angled  triangle  BHOi 


Now  by  Anal.  Geom.,  c2  —  a9  =  52. 
Therefore  2b  =  BH  =  the  minor  axis. 
Draw  the  chord  PB  and  produce  the  arc  AP  to  cut  it  in  0, 
Then  by  Geom.  (Table  I.  24) 

%H*  -  PB  X  OB  =  2R*  sin  i  A2  X  2(£2  -  J?i)  sin  i  A2 
.  •  .  J5//  =  2  sin  i  A  a  ^-^fBa  -  #1)  (137) 


COMPOUND   CURVES.  123 

Let  a  =  the  angle  HO^,  then 

tan  a  —  -Sj^-  and  BOj.  =  ^^  (138) 

In  the  triangle  BOM  let  O^BO*  =  /3  •  then 

sin  ft  =      ap<0    —  sin  A2  (139) 


The  polar  equation  of  the  hyperbola  for  the  branch  10M, 
taking  the  pole  at  B  and  estimating  the  variable  angle  v  from 
the  line  EOi,  is 


c  .  cos  v  —  a 

When  v  =  fi  ±  i,  r  —  R*,  and  substituting  the  values  of 
a,  b,  and  c  found  above,  we  have 

~DTT$ 


*       2  (BO,  cos  (/?  ±  t)  -  ^0 

using  (yS  -|-  *')  when  F'  falls  between  F  and  J.,  as  in  the 
figure,  and  (fi  —  i)  when  V  falls  beyond  F. 

In  the  triangle  BOM,  the  angle  BO^'Oi  =  A2'  and 

50 
sin  A  a'  =  -p-r-  -^  sin  (/?  ±  *)  (141) 

±iz    —  ±i\ 

Finally 

P^P'  =  A,  -  (A,'  ±  »)  (142) 

Remark,—  When  F'  falls  between  Fand  ^4,  as  in  Fig.  52,  if 
the  angle  i  be  greater  than  the  angle  VBE,  the  curve  ceases  to 
be  a  compound,  and  becomes  reversed.  Therefore  VBH  = 
a  —  ft  is  the  maximum  value  of  /  possible  in  this  case.  When 
F'  falls  beyond  F,  the  point  P'  will  fall  between  Pand  A; 
and  the  largest  possible  value  of  I  will  then  be  that  which 
renders  POiP'=  AI,  and  makes  the  point  P'  coincide  with  A 


124 


FIELD   ENGINEERING. 


Example.—  Fig.  52.     Let  Rl  =  1432.69 
i  =6°        jR2  =  2292.01 
(137)  R*  -  R!      859.32 
Rt    2292.01 


(138) 


2 
BH 

a 
BO, 


1432.69 


(139)  £2  - 


A2 


(140) 


28° 


42°  36'  23". 7 
42°  36'  23".7 


56° 

21°  28'  06".3 
27°  28'  06".3 


1727.09 
1432.69 


BH* 
J?2' 


(141)  .-. 

(142)  .*. 


A  Q1  ° 

Ai    =   dl 

A2  =  56° 

log  2.934155 
3.360217 


2  )  6.294372 

3T47186 

log  sin  9.671609 

0.301030 

3Tl9825 
3.156151 

log  tan  9T963674 

log  cos  9.866889 

3.289262 
2.934155 


9.644893 
log  sin  9.918574 

log  sin  9^563467 

log  cos  9.948053 

3.289262 


294.40  X  2  =  588.80 

2949.05 

A  a'  =  36°  18' 26" 
POiP'  =  13°  41'  34"  =  342.3  feet. 


3.237315 


2.769968 
6.239650 


3.469682 


Remark — This  problem  may  also  be  solved  by  first  finding 
the  new  sides  VA,  V'B,  from  which  and  the  new  central 
angle  (A  ±  i),  and  the  radius  Ri,  may  be  found  AI',  Aa',  and 
Rz,  as  in  §  162.  The  new  sides  are  readily  found  from  the 
old  ones  by  solving  the  triangle  VBV.  If  the  original  sides 
are  not  given,  they  must  be  calculated  as  in  §  164. 

173.  Given:  a  compound  curve  ending  in  a  tangent,  the 
last  radius  being  the  less,  to  change  the  last  radius  and  the 
position  of  tlie  P.C.C.  so  that  the  curve  may  end  at  the  same 
tangent  point,  but  with  a  given  difference  in  the 
direction  of  tangent.  Fig.  53. 


COMPOUND   CURVES.  125 

Let  APB  be  the  given  curve,  and  P02  =  #2,  and  P0j  = 
Ei  <  E-i.  Let  F'J5be  the  new  tangent,  and  VBV  =  i,  the 
given  angle;  to  find  BOi  =  Hi,  BOt'P'  =  A/,  and  P02P'. 

We  have 

BO,  +  Ot02  =  &  -f  CRa  -  JBO  =  -Ra 

P(V  +  Oi'Oa  =  £/  -f  (#„  -  J2/)  =  -Ra 

from  which  we  infer  that  the  locus  of  the  centre  Oi  is  an 
ellipse,  of  which  B  and  02  are  the  foci,  and  R*  the  major  axis, 


FIG.  53. 

since  the  sum  of  the  distances  SOi  and  020i'  is  always  equal 
to  P2. 

This  suggests  an  easy  graphical  solution  of  the  prob- 
blem,  as  follows  : 

Perpendicular  to  V'B  draw  the  indefinite  line  BK,  which 
will  contain  the  required  centre  O/,  and  layoff  BK  =  i?2. 
Join  KOi,  bisect  it,  and  from  the  middle  point  erect  a  perpen- 
dicular to  intersect  BK  in  0,'.  Join  02O,',  and  produce  the 
line  to  intersect  the  arc  AP  (produced  if  necessary)  in  P', 
which  is  the  new  P.  C.C.  required.  P'O/  =  SOi  =  J?i',  the 
required  radius,  and  P'O^B  =  AI'. 

The  analytical  solution  is  as  follows  :  Adopting  the 
usual  notation  of  the  ellipse, 

let  2a  —  Hi    ,=  the  major  axis, 
"  2c  =  P02  =  the  distance  between  foci. 
At  B  erect  5J3"  perpendicular  to  BO?  to  intersect  the  arc  AP 


126  FIELD 

(produced  if  necessary)  in  E,   and  join  HO*  =  B*.     Then 

BE'2  =  RJ  -  BOZ°-  =  ±a?  -  4c2 
But  by  Anal.  Gebm.,  «2  -  c2  =  62. 
Hence  2b  —  BE  =  the  minor  axis. 

In  the  triangle  B0t02  we  know  BO,  —  Rt,  and  0,0*  = 
R-i  —  Hi,  and  the  included  angle  jB0j02  =  180°  —  A,  ;  hence 
by  Trig.  (Tab.  II.  25) 

27?          7? 

tan  1(0!  0,B  -  O^BOJ  =  -  *       **  tan  |  A  ,         (143) 

Ji2 

The  angles  at  B  and  02  are  then  found  by  (Tab.  II.  26). 
Let  ft  =  the  angle  0,BO^ ;  then 

m  =  (ft-lZi)^V  (144) 

The  value  of  BE*  above  may  be  written 

BE*  =  (R*  +  B0.2)  (7?2  -  BOJ  (145) 

The  polar  equation  of  the  ellipse,  taking  the  pole  at  B,  and 
estimating  the  variable  angle  v  from  the  axis  BOZ,  is 


a  —  c  .  cos  v 


When  v  =  0  •?  i,  then  r  =  J2/f  and  substituting  the  values 
of  a,  b,  and  c,  given  above,  we  have 


O) 


using  (ft  —  i}  when  F'  falls  between  Fand  ^4,  as  in  Fig.  53, 
and  (/?+*)  when  F'  falls  beyond  F. 

In  the  triangle  BOi'Oi,  the  angle  Oi'503  =  (/?T  0,  and  the 
exterior  angle  BOi'P'  =  A/;  hence 

J50 

sin  Ai'  =  -„  --  5-7  sin  (/?  T  »)  (147) 

Ji2  —  *6 

Finally  PO,P'  =-(A,  T  a)  -  A/  (148) 

When  F'  is  on  AV,  then  PO^P'  is  negative,  showing  that 
it  must  be  laid  off  from  P  toward  A;  but  when  V  is  beyond 


COMPOUND   CURVES. 


12? 


F,  then  P02P'  is  positive,  and  P'  will  tie  on  AP  produced. 
The  only  limits  imposed  on  the  angle  i  are  that  the  resulting 
value  of  PP'  shall  not  exceed  PA,  and  that  Ri  shall  not  be 
less  than  a  practical  minimum. 

Example.— Fig.  53. 

Let  D9  =  3°  20'      R*  =  1719.12      A2  =  23°  20' 

D.  =  6°  R,=    955.37      A,  =  48°  *  =  7°  45' 


The  resulting  values  are  as  follows: 


ft 

BO, 
J3H* 


A/ 

O,P' 

PP' 


1572.42 
1273.65 


440.5 


21°  09'  32".6 


54°  56' 
14°  41' 


3.196567 
5.683829 
3.105052 


(See  also  remark  at  end  of  §  172.) 

174.  Given  a  simple  curve  joining  two  tangents,  to  re- 
place it  by  a  three-centred  compound  curve  between 
the  same  tangent  points.  Fig.  54. 


J50aP' 


Fid.  54. 

Let  R  =  AO  —  radius  of  simple  curve. 
Rv  =  P0l  =  P'0l  <R    Ai  =  POi 
Rv  =  AO*  =   J303  >  R    A3  =  AOi 

A  =  AOB        . 

Since  AO*  is  made  equal  to  BO*  and  VA—VB,  AOiPmust 
equal  BO$P',  and  the  compound  curve  will  be  symmetrical 
about  the  bisecting  line  V0\  and  the  centre  Oi  will  be  on  the 
line  VO. 
We  have  at  once  from  the  figure, 

2A2+  A,  =   A  (149) 


128 


FIELD   ENGINEERING. 


In  the  triangle  00i02  we  have 

Oi02  :    00-i  ::  sin  AOV:  sin 
whence 


(150) 


which  expresses  the  general  relation  between  the  quantities, 
R  and  A  being  given. 

We  may  now  assume  values  for  Ri  and  J?2  subject  to  the 
above  conditions,  viz.,  -Bi  <  R  and  Ri  >  R;  whence 


sin  i  A  i  = 


t  —  R)  sin 


(151) 


In  selecting  values  for  R^  and  Ri,  the  degree  of  curve  DI 
should  be  but  little  greater  than  D  of  the  simple  curve,  say 
from  30  to  60  minutes,  while  i>2  may  be  taken  at  i-D  to  ±D. 

'.-Given:      R  =  1719.12    D  -  3°  20'     A  =  40° 


=  1432.69    Dl  =  4° 
,  =  5729.65    Z>2  =  la 


-  R 


4010.53 
4296.96 


4  A 


Ai 

AP  =  P'B   138.4  ft. 


20' 

18°  36'  57" 

37°  13'  54" 

1°  23'  03" 


log  3.603202 
"  3.633161 

"  9^970041 
log  sin  9.534052 

"  "  9.504093 


Again  we  may  assume  A2  and  Ri,  whence 

Ai  =   A  —  2A3 
and 

_  gsinjA  -  Ri 
sin    A  —  sin 


(152) 


Example.—  Given:  R  =  1719.12     A  =  40° 

Let  Rl  =  1432.69     A2  =  1°  .'.    Ai  =  38° 
Am.  R*  =  7387.24     .  •.  Da  =  0°  46f    AP  =  129. 

Finally  we  may  assume  A  3  aw-^  .#2,  and  deduce  A  i  and  Ri 
from  eqs.  (149)  (150);  but  this  is  the  least  desirable  because 


COMPOUND   CURVES.  129 

the  valug>  of  Ei  so  found  will  not  usually  give  a  convenient 
value  to  the  degree  of  curve  Di. 

175.   To  determine  the  distance  HH'  between  the  middle 
points  of  a  simple  curve  and  a  three-centred  compound  curve 
joining  the  same  tangent  points  AB.    Fig.  54. 
In  the  triangle  OOi  02,  we  have 


>  =     a  -  .        - 

sin  i  A 
HH'  =  001  -f  0,H'  -  OH 

...  HH'  =  (Z?2  -  *i)*  -  (5  -  *0      (153) 


In  the  first  example  given  above  HH'  —  14.55,  and  in  the 
second  HH'  -  17.05  ft. 

In  many  instances  the  distance  HH'  is  so  great  as  to  render 
this  problem  practically  useless,  unless  the  distance  HHi  is 
discounted  beforehand  by  putting  the  simple  curve  AHB  a 
sufficient  distance  inside  of  the  proper  location  through  the 
point  H'.  But  the  problem  given  below  is  usually  preferable. 

176.  Given,  a  simple  curve  joining  two  tangents  to  re- 
place it  by  a  three-centred  Compound  curve  which 
shall  pass  through  the  same  middle  point  H. 

I.  The  curve  flattened  at  the  tangents.    Fig.  55. 


=  AO,  the  radius,  and  A  =  the  central  angle  of  the 
simple  curve  AHB,  and  let  H  be  the  middle  point. 

Let  ^  =  PO,  =  HO,  A  ,  =  POiP' 


"  A'  and  B'  be  the  new  tangent  points  required. 
We  have  at  once,  as  in  the  last  problem, 

2A2+A1  =  A.  (154) 


130 


FIELD 


Since  the  curve  is  to  be  symmetrical  about  VO,  HM  =  HP'. 
PA  =  P'B,  and  A  A'  =  SB'. 


In  the  diagram  produce  the  arc  HP  to  G,  and  draw  Oi  0 
parallel  to  OA,  and  produce  it  to  K.  Then  a  tangent  line  al 
G  will  be  parallel  to  VA\  and  by  §  137  the  point  G  will  be  OD 
the  long  chord  HA,  and  on  the  long  chord  PA'.  GK  is  the 
perpendicular  distance  between  parallel  tangents,  and  the 
problem  is  similar  to  that  given  in  §  171  ;  whence  by  eq.  (57) 
we  have,  in  this  case, 

OK  =  (Ba  -  BO  vers  A  2  =  (B  —  .BO  vers  i-  A  .     (155) 

for  the  general  equation  in  which  B  and  A  are  given. 
Analagous  to  eq.  (130)  we  have 


AA'  =  KA  -  KA  =  GK  cot  GA'K  -  GKcot  GAK. 


.  '  .  AA'  =  GK  (cot  |  A  3  -  cot  i  A) 


(156) 


in  which  GK  is  obtained  from  (155). 

We  may  now  assume  values  for  Bi  and  B2,  making  Bi  <  B 
and  B2  >  B,  and  deduce  the  values  of  A2>  Ai,  and  AA. 

Solving  eq.  (155) 


(B  -  BQ  vers  JA  _ 
vers  A  a  =  -  5  -  D  -  = 


Eq.  (154)  gives  AI,  and  eq.  (156)  gives  AA'. 


COMPOUND   CURVES.  131 


Example.— Fig.  55. 

Given:  R  =    764.489  D  =  7°  30'         A  =  40° 

Let^  =    716.779  D,  =  8° 

"  R2  =  3437.870  2?3  =  1°  40' 

(155)  R-  R,          47.71  log  1.678609 

|  A  20°  log  vers  8.780370 

GK  log  0.458979 

J22  -  Rl       2721.091  "  3.434743 

A  2  (say)  2°  38'        log  vers  7.024236 

A'P        158.00        Ax  =34°  44' 

(156)  iA2          43.5081   =  cot    1°  19' 

±A  5.6713        cot  10° 

37.8368  log  1.577914 

OK  "  0.458979 


A  A        108.87  "  2.036893 

Again,  we  may  assume  A  2  and  Ri  <  R\  whence 

Ai  =   A  —  2A2 

and 

eq.  (155)     GK  =  (R  -  RJ  vers  i  A 
and 


Eq.  (156)  gives  AJ.'. 

Again,  we  may  assume   A  2  and  the  distance  A  A  ;  whence, 
from  eq.  (156) 

AA> 


eq.  (155)       R,  =  R  - 


--  — 

—  COtiA 
nir 


(159) 


vers 


eq.  (158)  gives  R*. 
Again,  we  may  assume  &  <  R  and  AA  ;  then,  eq.  (155) 

GK=  (R-  iJJOversiA 
and  eq.  (156) 

A   A' 

cot  i  Aa  =  cot  i  A  +  ~  (160) 

and  eq.  (158)  gives  R*. 


132 


FIELD 


Examjde. — Fig  55. 

Given :  R  =  764.489 
Let  jKi  =  716.779 
"  A  A  =  110. 

Hence  by  last  example, 


D  =  7°  30'        A  =  40° 


GK 
eq.  (160)  AA 

110. 

-*v                              log  0.458979 
2.041393 

(158) 

COtiA 
COt  |A2 
A2 

GK 

xl 

AP' 

38.2309 
5.6713 

1.582414 
10° 

1°  18'  18"       log  1.642486 

43.9022 

2759.5 
3476.3 
157. 

(say)      2°  37'      log  vers  7.018147 
0.458979 

3.440832 
Da  =    1°  39' 

Ai  =  34°  46' 

II.  The  curve  sharpened  at  the  tangents.    Fig.  56. 

This  case  will  only  occur  when,  with  a  given  external  dis- 
tance VH,  a  simple  curve  would  absorb  too  much  of  the  tan- 
gents. 


FIG.  56. 


Let  AHB  be  the  simple  curve,  and 
"   A'PHP'B'the  required  compound  curve 


We  have  from  the  figure, 
2A4 


A2= 

O^  A,  = 

Aa=  A. 


(161) 


COMPOUND   CUEVES.  133 

In  the  diagram  draw  0?G  parallel  to  OA  cutting  the  tan- 
gent at  K,  and  produce  the  arc  HP  to  O.  Draw  the  chords 
QH  and  GP,  passing  through  A  and  A'  respectively.  We 
have  then  a  discussion  similar  to  the  preceding  case,  and  to 
the  problem  §  171,  Fig.  51,  whence  we  derive  the  general 
f  ormula?  : 

GK  =  (R*  -  R,)  vers  A  i  '  =  (It*  -  R)  vers  i  A      (162) 
and 

AA  =  GK(cot  iAi  —  cot  iA)  (163) 

1.  Assuming  Ri  <  R  and  R*  >  R 

vers  Aj  =  R^  _^  =   ^~^  vers  |A         (164) 

2.  Assuming  Ai  <  iA  and  R!  <  R 

R  vers  j  A  -  R*  vers  AI 

vers  i  A  -vers  A: 

3.  Assuming  AJ  <  iA  and  A  A 

'    e*=  (166) 


(167) 


1    vers  |  A 

tfx  =  R,  --  °^—  (168) 

vers  AI 

4.  Assuming  Ry  >  R  and  -4.4' 

(?.?=  (,R2  -  R)  vers  i  A 
j  AI 

cot  |  A  ,  =  cot  i  A  +  -  (169) 


The  third  assumption  will  usually  secure  most  readily  the 
desired  curve.  AA  should  be  assumed  as  small  as  the  nature 
of  the  case  will  allow,  and  AI  should  not  be  much  smaller 
than  £  A  . 

It  is  evidently  not  necessary  that  the  new  curve  should  be 
symmetrical;  for  having  laid  out  the  curve  A'PH,  the  simple 
curve  HB  may  then  be  used,  or,  if  desirable,  some  compound 
curve  HP'B'  determined  by  an  assumed  value  of  BB'  not 
equal  to  AA. 


134  FIELD   ENGINEERING. 

These  formulae  (154)  to  (169)  are  readily  adapted  to  the 
case  of  substituting  a  compound  for  a  simple  curve  when  it 
is  necessary  to  keep  one  tangent  point  fixed,  but  to  move  the 
other  a  certain  distance  in  either  direction  on  the  tangent. 
For  if  in  Figs.  55,  56,  we  draw  a  tangent  at  H,  and  make  // 
the  fixed  point  of  tangent,  it  is  evident  that  the  central  angle 
of  the  curve  will  then  be  AOH.  The  only  change  necessary, 
therefore,  to  adopt  the  formulas  to  this  case  is  to  write  A  in 
place  of  i  A,  and  to  observe,  instead  of  eqs.  (154)  (161),  that 

Aj  -j-  A2  =   A. 

Example. — Fig.  55. 

Let  11  =  1910.08     A  =  84° 

Assume    AA  =     260.         AI  =  38°     .'.  A2  =  8° 

Eq.  (166)  AA'  =    260.  log  2.414973 

cot  i  AH  2.90421          19* 

cotiA  2.60509          21° 


.29912  log          9.475846 

GK  "           2.939127 

Eq.    (167)  i  A  42°                "  vers  9.409688 

3384.07  3.529439 
E        1910.08 

R*        5294.15  D  =  say  1°  05' 

Eq.  (168)  GK  log          2.939127 

Ai  38°                "  vers  9.326314 


4100.27  3.612813 


Rl        1193.88  D  =  4°48' 

A'P         791.67  PS  =369. 23 

177.  Given,  two  curves  joined  by  a  common  tangent 
to  replace  the  tangent  by  a  curve  compounded  with 
the  given  curves.  Fig.  57. 

Let  EI  =  BOi  the  radius  of  one  curve, 
"    Rs  =  A03  the  radius  of  the  other  curve,  >  jf?i, 

I  —  BA  the  common  tangent, 

"    J?2  =  POi  =  P'02  the  radius  of  connecting  curve. 
"    A2  =  P#2P'  the  central  angle  of        " 
"      a=  AOSP'  and  /S  =  BO^P. 


COMPOUND   CURYES. 


135 


In  the  diagram  join  Oi03  and  draw  OiG  parallel  to  BA. 
Then  in  the  right-angled  triangle  OiGOa  we  have, 


0,0,  = 


R3  —  RJ 

cos  i 


(170) 


sin* 


which  gives  the  distance  between  the  centres  of  the  given 
curves. 


FIG.  57. 

We  shall  now  assume  the  following  geometrical  truths, 
which  may  be  easily  demonstrated. 

If  two  circles  intersect  in  one  point,  they  intersect  in  two 
points;  and  the  line  joining  the  two  points  is  the  common 
chord. 

The  common  chord  is  perpendicular  to  the  line  joining  the 
centres,  and  when  produced  it  bisects  the  common  tangents. 

If  a  third  circle  is  drawn  touching  the  two  circles,  a  tangent 
to  the  third  circle,  parallel  to  the  common  tangent,  will  have 
its  tangent  point  on  the  common  chord  produced. 

Conversely,  therefore,  if  the  tangent  BA  be  bisected  at  K, 
and  a  line,  KI,  drawn  perpendicular  to  0i#8,  KI  will  coincide 
with  the  common  chord  produced,  and  the  angle  IKA  = 
A030i  =  i.  If  on  KI  we  assume  a  point  /  through  which 
it  is  desirable  that  the  connecting  curve  should  pass,  then  /  is 
the  tangent  point  of  a  tangent  parallel  to  BA ;  consequently 
a  line  through  /  perpendicular  to  BA  contains  the  required 
centre  Ot. 


136  FIELD   ENGINEERING. 

I.  Let  p  —  HI  =  the  perpendicular  distance  between  the 
tangents. 

If  in  the  diagram  we  join  IA  and  IB,  and  produce  the 
chords  to  intersect  the  given  curves  in  Pand  P',  then  Pand 
P'  are  the  points  of  compound  curvature;  and  the  lines  P0t 
and  P  03  produced  will  intersect  /02  in  the  same  point  02; 
and  the  angles  P'02/  =  a  and  PO27  =  /?. 

In  the  triangle  AIB  the  line  KI  bisects  the  base  AB,  and 
we  have  by  Geom.  Tab.  I.  25. 

AIn-  +  P/2  =  2AK*  -f  2KI* 
By  eq.  (56)         AI  =  2(#2  -  R9)  sin  ia 
BI  =  2(I?a  —  -BO  sin  $/3 

AK  =  #     and     KI  =  -^-^ 
Bint 


Dividing  by  2  and  putting  vers  a  =  2  sin2  £a  and  vers  ft  = 
2  sin2  i/3  (Tab.  II.  46) 

(5,  -  P3)2  vers  a  +  (P2  -  J20»  vers  /?  =  *P  +  ^ 
But  by  eq  (57) 

(i?3  -  JB,)  vers  a  =  (JSa  —  J?,)  vers  /J  =  ^         (172) 


(173) 

tJJ  Dill      I/ 

From  (172) 

vers  a  =       ^_       ;  vers  fi  =       ^_  (174) 

and  from  the  figure 

As  =  a  -j-  ft  (175) 

These  forrmilae  solve  the  problem  when  p  is  assumed.    If 
desirable  we  may  find  a  and  ft  independently  of  R*,  for  in 


COMPOUND   CUEVES.  137 

the  triangle  AIB,   IAB  =  $a  and  ISA  =  \fi\   and  since 
HK  =  p  cot  i, 

A  TT  17  TTTr  J 

(176) 
(177) 


II.  In  case  a  or  ft  is  assumed,  we  have  from  the  last  equa- 
tion 

P  =  2(cot  |a  +  cot  0  =  2(cot  */J  -  cot  t) 


III.  Jtt  ease  ^e  radius  It*  is  assumed,  then  in  the  triangle 
01026>3  we  know  all  three  sides;  for  dOt  =  (R*  —  Mi), 

Oa03  =  (Sa  -  RJ,  and  Oi03  =  — ^— ^ 

COS  2 


By  Trig.  (Table  II.  31.) 
_ 


in  which  *  =  \  sum  of  the  three  sides. 

Substituting  values,  and  reducing,  observing  that, 


and  that  (#3  —  Mi)  tan  i  =  I,  we  have 

vers  A2  =  H7p  -  p^T-p  -  p-v  (179) 

4      — 

In  the  same  triangle. 

sin  0,0,0,  =  sin  A2 


But  from  the  figure  030,0^  =  i  -  ft,  and  taking  the  value 
of  0j03  from  eq.  (171). 


138  FIELD   ENGINEERING. 


sin  (i-(S)  =      .-.  (180) 

We  then  find  a  from  eq.  (175)  and  p  from  (172). 

The  angles  a  and  ft  may  be  found  otherwise,  for  by  Trig= 
(Tab.  II.  27)  we  have  in  the  triangle  0i0a03 

sin  K0i0,0«  -  0,0!  0.)  =  °l0*0~0°*°3  cos  iA3 
or 

sin  (90"  -  (i  +  5L=£A  =  (R-AXiOBicoBtA. 

\  a       I  Mz  —  M\ 

.  •  .  cos  li  -\  ---  ~-  1  =  cos  i  .  cos  £  AJ»  (181) 

which  is  a  convenient  formula  when  i  and    A  2  are  not  too 
small.     Having  obtained  —  ^-,  we  have 


For  a  constant  value  of  I  the  less  the  difference  of  E3  —  Ei 
the  greater  will  be  the  value  of  the  angle  i.  When  J?3  =  'Rlt 
cot  t  =  0  and  i  =  90°  and  the  tangent  point  /  will  be  on  a  per- 
pendicular to  BA  drawn  through  the  middle  point  K;  and 
a  =  ft.  On  the  contrary,  as  (R3  —  R^  increases,  i  becomes 
less,  and  the  foot,  H,  of  the  perpendicular  ///moves  toward 
B,  the  tangent  point  of  the  curve  of  smaller  radius  JRi.  The 
distance  HK  =  p  cot  i.  The  connecting  curve  is  farthest 
from  the  tangent  BA  at  /.  To  find  the  ordinate  from  BA  to 
the  curve  at  any  other  point,  subtract  from  p  the  tangent 
offset  for  the  length  of  curve  from  /  to  the  ordinate  in  ques- 
tion. §115,  eq.  (39)  may  be  used  on  flat  curves  with  tolera- 
ble accuracy,  even  when  the  distance  equals  several  hundred 
feet. 

IY.  It  is  evident  that  in  this  problem  R*  must  be  greater 
than  either  Hi  or  /?3.  As  the  centre  02  is  taken  nearer  the 


COMPOUND   CURVES. 


139 


line  0i  03,  M-2  grows  less,  and  is  a  minimum  when  02  falls  on 
the  line  Oi03.     In  this  case  we  have  A2  =  180°,  and 


-f  Rt  +0103);  a  minimum. 


(183) 


This  limit  must  be  regarded  in  assuming  the  value  of  M-,. 
Since 

0!02  -  0203  =  CR3  -  M,)  -  (Mi  -  R3)  =  (R3  -  R,) 

a  constant  value,  independent  of  R*,  we  infer  that  the  centre 
02  is  always  on  a  hyperbola  of  which  Oi  and  03  are  the  foci; 
(R3  —  M\)  equals  the  diameter  on  the  axis  joining  the  foci; 
and  I  equals  the  diameter  at  right  angles  to  it,  for  in  the  tri- 

angle OiGOs, 

I*  =  0^2  -  (R3  -7?i)2  (184) 

Example.  —  Fig.  57. 


Given  : 
Assume 
Eq.  (170)  R3 

Eq.  (173) 

R> 

-I 

I 
i 

i 
i 
P 
* 

P 

=  1432.69 
=      11.4 
477.39 
400. 

11.4 
27.64 

3508.77 
3342.77 

M3  =  1910.08  and  I-  400. 
to  find  R3,a  and/?, 
log  2.678873 
"    2.602060 

39°  57'  34"    log  cot  0.076813 

39°  57'  34"      "  sin 
39°  57'  34"      "  sin2 
log 

<  < 
« 
if 

9.807701 
9.615402 
1.056905 

1.441503 
4.602060 
1.056905 

3.545155 

Eq.  (174) 


p 
-  M, 

a 
P 


2)  6879.18 

3439.59  (say)  3437.87 

11.4 
1527.79 


1.056905 
3.184064 


11.4 
2005.18 


ft          (nearly) 


7°  00'     log  vers  7.872841 

log  1.056905 

"    3.302153 


6°  07'    log  vers  7. 754752 
13°  07' 


140 


FIELD 


Example.— Fig  57. 

Given:    ^  r^  1432.69,  R3  =  1910.08,  and  I  =  400. 
Assume  R*  =  3437.87,  to  find  A2,  ft,  a  and  p. 


Eq.  (179) 

A- A 
A  -  £3 


Eq.  (170)  A- 


Z2 

A3 
\ 


Eq.  (180) 


A2 

-  R* 


Eq.  (175)  a 

Eq.  (172)  £a  -A 


2. 

2005.18 
1527.79 


477.39 

400. 


1527.79 


400. 


13°  07'  22" 


39°  57'  34" 
39°  57'  34" 
13°  07'  22" 


33°  50'  39" 

6°  06'  55" 
7°  00'  27" 

7°  00'  27" 


log  0.301030 
"  3.302153 
"  3.184064 


"  6.787247 
5.204120 

log  vers  8.416873 

log  2.678873 

"  2.602060 


11.41 


log  cot  0.076813 

log  sin  9.807701 

"     "  9.356099 

log  3. 184064 

log  sin  2.347864 
log  2. 602060 

log  sin  9.745804 


log  3.184064 
log  vers  7.873309 

1.057373 


•  178.  Given:  a  three-centred  compound  curve  to  replace 
the  middle  arc  by  an  arc  of  different  radius. 

I.  When  the  radius  of  the  middle  arc  is  the  greatest. 
Fig.  57. 

First  find  the  length  and  direction  of  the  common  tangent 
AB.  Let  A  2  =  central  angle  of  the  middle  arc,  R*  =  its 
radius,  and  Hi  and  E3  the  radii  of  the  other  arcs.  From  eq. 
(179). 

I  =  V2(£2  -  J?0  (ft*  -  Rs)  vers  A,  (1 85) 

Then  find  i  by  eq.  (170),  a  and  /5  by  eqs.  (181)  (182),  and  p  by 
eq.  (172). 

For  the  new  arc  we  may  now  assume  a  new  value  for  p,  or 
for  Ri,  or  for  a.  Indicating  the  new  values  by  an  accent,  if 
we  assume  p'  we  proceed  as  in  the  last  problem,  using  eqs. 
(173),  etc.  If  we  assume  R*,  we  use  eq.  (179),  etc.  If  we 
assume  a',  we  use  eq.  (178). 


COMPOUND   CURVES. 


141 


II.  When  the  radius  of  the  middle  arc  is  the  least  of  the 
three.  Fig.  58. 

In  this  case  the  middle  arc  is  within  the  other  two  pro- 
duced; and  for- the  same  values  of  RiR3  and  Oi03,  the  locus 


FIG.  58. 

of  the  centre  02  is  the  opposite  branch  of  the  hyperbola  found 
in  §177.  When  the  centre  02  falls  on  the  line  Oi#3,  A2  = 
180°,  and 

Rz  =  K-R,  +  Mi  —  Oj  03),  a  maximum.         (186) 
Analogous  to  eq.  (185),  we  have 


I  =  V2(R1  -  R*}  (R3  -  J2a)  vers  A2  (187) 

which  gives  the  length  of  the  common  tangent  FZ. 

We  then  have  the  values  of  i  and  of  Ot03  by  eqs.  (170)  (171), 
and  of  a  and  ft  by  eqs.  (181)  (182),  and  analogous  to  eq.  (172), 

p  =  (^  _  Rs}  vers  a  =  (R*-R*)  vers  ft        (188) 

in  which  p  is  the  perpendicular  distance  HI  bet  ween  parallel 
tangents. 

For  the  new  arc  we  may  now  assume  a  new  value  for  p,  for 
.R2,  or  for  a.  Indicating  the  new  values  by  an  accent,  if  we 
assume  p',  we  have,  analogous  to  eq.  (173) 


FIELD   ENGIJSTEEKIKG. 


(189) 


and  from  eq.  (188) 

vers  a'  = 


;  vers  ft'  =  j^-~jr 

113  —  Jt2 


(190) 


If  we  assume  R2',  we  have,  analogous  to  eq.  (179), 

M 


vers  A 2   = 


(191) 


2(5!  -  52')  (53  -  58') 
and  we  find  a  and  /3  by  eqs.  (181)  (182),  and  p'  by  eq.  (188). 

III.   WJien  the  radius  of  the  middle  arc  has  an  intermedi 
ate  value,  compared  with  the  other  radii.     Fig.  59. 


FIG.  59. 
\ 

In  this  case,  whatever  be  the  value  of  J?2,  we  have 
0302  +  0^0,  =  (E3  -  R*}  -f  CR2  -  50  =  (B9  -  50 

a  constant  value  independent  of  52 ;  hence  we  infer  that  the 
locus  of  02  is  an  ellipse,  of  which  Oj  and  03  are  the  foci,  and 
(53  —  50  equal  to  the  transverse  axis. 

Let  I  =  QQ'  =  the  conjugate  axis,  and  let  i  =  Q030i  = 
Q0103. 

Produce   03Q  to   O,  making  QG  —  OsQ,  and  join  GO,. 


COMPOUND   CURVES.  143 

Then  by  similar  triangles  £0i  is  perpendicular  to  Oi03,  and 
O0t  =  I;  and  in  the  right-angled  triangle  (r030i 


O01 
sin  ^  =  -- 


Ol  03  =  (Rz  -  Ri)  cos  i  -  I  cot  t  (193) 

Analogous  to  eqs.  (185)  and  (187),  we  have 

I  =  V  2(R*  -  R*)  (R*  -  Ri)  vers  'A  ,  (194) 

•which  may  also  be  derived  from  the  triangles  0i0a03  and 
O&Q. 

Let  a  =  02030i,  and  ft  =  020x03 
Then 

sin  a  =  %%-  sin  A2  =  ^=^  tan  ».  sin  A2          (195) 
O\U» 

From  the  figure  ^  =  A2  —  a  (196) 

In  the  diagram  produce  the  line  030i  and  it  will  intersect 
all  the  arcs.  At  the  points  Z  and  T,  where  it  cuts  the  inner 
and  outer  arcs,  draw  tangent  lines  perpendicular  to  030i. 
Draw  the  radius  OaJ  parallel  to  0»0i,  and  the  tangent  line 
IL  at  I. 

Let  g  =  ZYauAp  =  ZL  =  HI 

Then  by  the  theory  of  parallel  tangents,  §137,  the  point  J  is 
on  the  chord  PZ  produced,  and  it  is  also  on  the  chord  P'  Y-, 
and  we  have 

p  =  ZL  =  (R*  -  R1)  vers  ft  .  (197) 

q  -  p  =  LT  =  (R3  -  jR2)  vers  a  (198) 

and  q  equals  the  sum  of  these.  But  q  =  ZFis.the  shortest 
distance  between  the  inner  and  outer  arcs,  and  has  a  constant 
value  independent  of  1?2.  If  we  assume  R*  =  i(R3  +  Ri)  the 
centre  02  will  be  at  Q,  and  a  =  ft  =  i,and  p  =  ±q.  Making 
these  substitutions  above, 

g  =  (R3  -  RJ  vers  f.  (199) 

Also,  from  the  figure, 


144  FIELD   ENGINEERING. 

ZY  =  03Y-  0,Z-  0,0,, 
or, 

q  =  E3  -  R,  -  0,03.  (200) 

In  the  triangle  ZIT  we  have  by  Geom.  Tab.  I.  26, 

ZP  =  IT*  +  ZY2  -  2ZY(ZY  -  ZL) 
or 

ZY2  -  2ZY.ZL  = 
Now, 


ZI*  —  4(^2  —  -fti)2  sin2 1/3  =  2(E*  —  E^  vers  /3 
1Y*  =  4(E3  —  -R2)2  sin2  la  =  2(Jfts  —  E*)  vers  a 

Hence 

ZP  =  2(^2  —  Ej.)  p  and  JF2  =  2(E3—  E*)  (q  —  p) 
Substituting  these  values,  and  solving  for  p,  we  have 

p  _  q(E3  -  E^-lg)  =  q(E3  -  J?2  -  lq)  ^^ 

Also 

E*  =  (E3  -  lq)  -  p  .  —^  (202) 

For  any  other  value  of  E^,  we  have 


Hence 

E*'  -  jR2  =  -~-?  (p  -  p')  (203) 

which  gives  the  change  in  R2  for  a  given  change  in  the  value 
of  p 

Observe  that  as  p  diminishes  7?2  increases  and  vice  versa. 

Having  determined  the  value  of  E*,  we  find  p'  by  substitut- 
ing E<t'  forE-2  in  eq.  (201);  and  from  eqs.  (197)  (198)  we  have 


vers  a'  =       ",  (205) 


COMPOUND   CURVES.  145 

and  the  change  in  the  points  of  compound  curvature  is  found 
by  (ft  -  ft')  and  (a1  -a). 

Remark. — When  R2  =  i(R3  -j-  Ri),  A2  =  2£,  a  minimum, 
and  the  long  chord  PP'  is  perpendicular  to  Oi03.  When  R* 
is  greater  than  this,  a  is  greater  than  ft,  and  vice  versa.  What- 
ever be  the  value  of  R?,  the  long  chord  PP'  always  cuts  the 
Hue  Oi03  produced  in  the  same  point  8,  at  a  distance  from  Zof 

ZS  =  Rl  vers  *; 
or  from  Oi  of  OiS  =  Ri  cos  i. 

This  item  will  be  found  useful  in  solving  the  problem 
graphically. 

Example. 

Let  Ri  =    781.84    A  =  7°  20' 
"     ft2  =  1375.40    J>2  =  4°  10'        A2  =  48° 
"    R3  =  1910.08    Z>3  =  3°  00' 
Let  p—p'=      11.30 

Eq.  (194)  2  log  0.301030 

R3  -  R2         534.68  "  2.728094 

R3  -  Ri         593.56  "  2.773465 

A  8  48°  log  vers  9.519657 

•  2)  5^322246 

I        458.27  log  2^661123 

(192)       R3  -  Rl       1128.24  "   3.052402 

i  23°  57'  55"  log  sin  9J608721 


(193)        i  23°  57'  55"  log  cos  9.  960847 

R3  -  Ri  log  3.052402 

0,0,  1030.98              log  *3.  013249 

(195)  ^2  -  R,  log  2.773465 

A  2  48°      log  sin  9.871073 

log  *  2^644538 

«  25°  19'  52"  log  sin  9.  631289 

(196)  ft  22°  40'  08" 

(203)              Oj03  log  3.01  3249 

(200)                    q  97.26                                            1.987934 


q 

p-p'  11.30               log  1.053078 

2'  -  R*  119.78               "  2.078393 

R*  1495^18  (say)  1494.95  for  3°  50'  curve. 


146  FIELD   ENGINEERING. 

(201)  7?3  -  RS  -  \q    366.50  log  2.564074 

—  "   1.025315 

q  

p'  34.57  "  1.538759 

(197)  .K2'  -  .Si  713.11  "  2.853157 

/?'  17°  55'       log  vers8. 685602 

(198)  q-p'  62.69  log  1.797198 
E3  -  Ri  415.13  "  2.618184 


31°  54'      log  vers  9. 179014 


Aa'  49°  49' 

a'  _  a  =  6°  34'  .-.  P'P"  =  218.89 
ft  -  ft'  —  ¥  45'   .-.  PP"     =    64.77 

The  practical  difficulty  in  changing  the  middle  arc  of  three 
centred  curves  lies  in  the  difference  of  measurement  that 
ensues.  Thus,  in  the  last  problem,  although  the  total  central 
angle  is  the  same,  the  new  curve  is  6.56  feet  shorter  than  the 
original,  making  a  fractional  station  at  P  ".  If  the  change  is 
made  during  the  location,  it  is  well  to  re-run  the  last  arc  from 
P"  to  the  tangent  following,  so  as  to  eliminate  the  fractional 
station  from  the  curve. 


TURNOUTS.  147 


CHAPTER  VII. 

TURNOUTS. 

179.  A  turnout  is  a  curved  track  by  which  a  car  may 
leave  the  main  track  for  another.  At  the  point  where  the 
outer  rail  of  the  turnout  crosses  the  rail  of  the  main  track  a 
frog  is  introduced  which  allows  the  flanges  of  the  wheels  to 
pass  the  rails.  A  frog  consists  essentially  of  a  solid  block  of 
iron  or  steel  having  two  straight  channels  crossing  each  other 
on  the  upper  surface,  in  which  the  flanges  of 
the  wheels  pass.  The  triangular  portion  of  the 
upper  surface  formed  by  the  channels  is  called 
the  tongue  of  the  frog,  and  the  angle  which  the 
channels  make  with  each  other  is  called  the  frog- 
angle.  Every  railroad  is  provided  with  a  set  of 
frogs  of  different  angles,  from  which  may  be 
selected  one  best  adapted  to  any  particular  case. 

The  frogs  may  be  designated  by  their  angles, 
but  it  is  customary  to  designate  them  by  numbers  expressing 
the  ratio  of  the  bisecting  line  FC  of  the  tongue  to  the  base 
line  ab,  Fig.  60.     Observe  that  F  is  at  the  intersection  of  the 
edges  produced,  and  not  at  the  blunt  point  of  the  tongue. 

In  the  triangle  aFC, 

•ryf 

' 


and  if  we  let  n  =  the  number  of  the  frog,  and  F  =  the  frog 
angle,  then 


On  some  roads,  however,  the  frogs  are  numbered  arbitrarily, 
or  according  to  their  length  in  feet,  while  on  others  they  are 
designated  by  letters  of  the  alphabet.  In  any  case  the  true 
number  (n)  of  a  frog  may  be  determined  by  the  above  for- 
mula. 


148 


FIELD   ENGINEERING. 


The  first  rail  of  the  turnout  is  common  to  both  tracks,  and 
is  called  the  switch-rail.  It  has  one  end  free,  so  as  to  be  shift- 
ed from  one  track  to  the  other  as  required ;  the  free  end,  D 
(Fig.  61),  is  called  the  point  of  switch.  The  tangent  point  of 
the  turnout,  at  A,  is  called  the  heel  of  switch,  and  the  distance, 
AD,  is  the  length  of  switch.  The  switch-rail  should  be  several 
feet  longer  than  AD,  and  the  excess  be  spiked  down  in  the 
line  of  the  main  track  back  of  the  point  A.  Then  if  the  point 
D  is  thrown  over  to  meet  the  rail  of  the  turnout  at  K,  the  switch 
rail  is  sprung  into  an  arc,  which  coincides  with  the  arc  of  the 
turnout,  provided  that  the  length  of  switch  AD  has  been  prop- 
erly taken.  The  distance  DK  through  which  the  point  moves 
is  called  the  throw  of  the  switch.  It  varies  on  different  roads 
from  4|  to  6  inches,  but  is  usually  made  about  5  inches,  or  0.42 
feet.  A  turnout  should  be  a  simple  curve  from  the  heel  of  the 
switch  to  the  point  of  the  frog. 

18O.  Owen:  a  main  track,  straight,  and  a  frog  angle  F,  to 
determine  the  distance  BF,  on  the  main  track  from  the  heel  of 
switch  to  point  of  frog,  the  radius,  r,  of  the  centre  line  of  the  turn- 
out, the  length  of  chord  af,  and  the  proper  length  of  switch  AD. 
Fig.  61. 


FIG.  61. 

Let     C  be  the  centre  of  the  turnout. 
"      F  =  the  frog  angle,  HFI  =  FCB. 
"      g  —  the  gauge  of  track  AB. 
"       r  =  radius,  aC  =  fC. 
"  DK  =  the  throw  of  switch. 

Then  the  radius  of  the  gauge  side  of  the  outer  rail  is  (r  -f-  ^g), 
and  we  have 


TURNOUTS.  149 

AB  =  FC  .  vers  FOB 
or, 

9  =  (r  +  iff)  vers  F 
whence 

;;";...;  '•"'-"  '.''*+*>  =  *£*  ;         (207) 

The  angle 


and         jB^1  —  AB  cot  J.JPB  =  #  .  cot  \F  (208) 

Again,  in  the  triangle  FCB 

BF=FC  .  sin  FCB  =  (r  +  &)  sin  ^  (209) 

The  chord  of  is,  evidently 

of  =  2r  sin  \F  (210) 

• 
Similar  to  eq.  (207),  we  have 

DK         DK 
vers  ACD  =  -j^r  =  - 

-S^C7        r  +  4# 

But  since  the  inside  rail  has  the  same  throw,  while  its  radius 
is  (r  —  \g),  we  may,  if  convenient,  drop  the  \g>  and  hence  the 
length  of  switch  is 

AD  =  r.  sin  ACD  (211) 

The  degree  of  curve  corresponding  to  r  is  found  from  Table 
IV.,  or  by  eq.  (17),  and  the  centre  line  of  the  turnout  may  be 
located  by  transit  deflections  from  the  tangent  point  a,  using 
chords  of  20  or  25  feet  -|-  the  correction  found  in  §§  106,  107; 
or  the  deflection  for  a  20-foot  chord  may  be  calculated  at  once 

by 

sin  (tf  „)  =  -^  (212) 

181.  Simple  as  these  formulae  are,  they  may  be  rendered 
still  more  convenient  by  introducing  the  number  of  the 
frog:,  n.  By  eq.  (206)  we  have  cot  \F  —  2n,  which  substi- 
tuted in  eq.  (208)  gives 

BF-  2gn  (213) 

Drawing  the  chord  AF  to  the  outer  rail, 


AF  =   V  AB*  -f  BF*  =  gVl  +4n*  (214) 


150  FIELD 


Make  BA'  =  AB  and  join  FA'  ;  then  by  similar  triangles, 
AA'F  and  AFC, 

AA  :  AF  ::  AF  :  FO 
whence 

AF* 

FG  =  4  ~ 
AA 

or  (r  +  ft)  =  \g  (1  -f  4^)  (215) 

whence  r  =  2gn*  =  BF  .  n  (216) 


The  chord  af  to  the  arc  of  the  centre  line  is  to  AF  as  r  is  to 
(f  +  iff} ;  hence  af  =  - 
eqs.  (214)  (215)  we  have 


;  hence  «/ =  — ~I~>  an(*  substituting  values  from 


(217) 


4/1  -f  4»* 

Assuming  that,  for  small  angles,  the  tangent  offsets  vary  as 
the  squares  of  their  distances  from  the  tangent  point,  which 
will  lead  to  no  material  error  in  this  case; 


whence  AD= 


AB:DK::  BF'2  :  AD* 
DK 


AB  V          (218) 


or  AD  =  V±n*.DK--    V2r  .  DK 


It  is  not  necessary  to  determine  the  degree  of  curve  in  order 
to  locate  the  turnout,  for  having  fixed  the  position  of  BF,  the 
position  of  af  is  found  by  laying  off  Ba,  and  F/,  each  equal  to 
iff.  Whatever  be  the  length  of  the  chord  af,  found  by  eq. 
(217)  or  (210),  its  middle  ordinate  is  always  ±g,  and  the  ordin- 
nates  at  the  quarter  points,  f  .  \g  —  ^g.  Thus  for  the  stan- 
dard gauge  of  4.708  the  middle  ordinate  is  1.177,  and  the  side 
ordinates  0.883. 

By  the  preceding  formulae  Table  XI.  has  been  calculated, 
which  gives  the  required  parts  of  a  turnout  for  various  frogs 
when  the  gauge  is  4  feet  8|-  inches  and  the  throw  5  inches; 
also  for  a  gauge  of  3  feet  and  throw  of  4  inches.  For  any 
other  throw,  only  AD  must  be  calculated.  For  a  different 
gauge  the  engineer  will  do  well  to  construct  a  similar  table, 
adapted  to  the  frogs  used  on  the  road. 


TURNOUTS. 


151 


In  the  table  the  frog  angle  is  given  to  seconds,  in  order  that 
the  results  may  agree,  whether  found  by  equations  in  §180 
or  §181;  but  in  practice  the  nearest  minute  is  sufficiently 
exact.  The  frogs  most  used  for  single  turnouts  are  those 
from  No.  7  to  No.  9,  inclusive. 

182.  In  case  of  a  double  turnout  from  the  same  switch, 
three  frogs  are  required,  as  at  F,F'  and  F",  Fig.  62.,  and  the 


switch  is  called  a  three-throw  switch,  because  its  point  takes 
three  positions.  The  frogs  F  and  F1  are  usually  alike,  and 
placed  exactly  opposite  each  other  in  the  main  track.  .  The 
other  frog  F"  is  placed  on  the  centre  line  of  the  main  track. 
Its  angle  F"  and  its  distance  from  a  are  now  to  be  determined 
in  terms  of  F. 

In  the  figure  we  have  vers  F"Ca  =  -^7-,-^  or 


The  distance 
also 


aF"  ~(r  +  i?)  sin 

aF"  =  r  .  tan  \~F" 


(219) 

(220) 
(221) 


All  the  parts  of  the  turnout  required  to  locate  the  frogs  F 
and  F"  are  calculated  by  the  formulae  in  the  preceding  sec- 
tions, or  are  taken  from  Table  XI. 

If  we  let  n"  =  the  number  of  the  frog  F",  then  by  eq.(206) 

tan  \F"  =  -j-pr,  which  substituted  in  eq.  (221)  gives 


aF"  = 


2n" 


(222) 


152  FIELD  EKGItfEEKING. 

Also,  in  the  triangle  aF"C, 

Equating  these  and  replacing  r  by  Zgn*,  we  obtain 


If  we  neglect  the  i,  we  have 


(approx.) 


n"  =  -11.  =  .  707171 


(233) 


(224) 


(225) 


Example.— It  F  —  F'  =  6°  44',  or  ra  =  n'  =  8.5,  then  n"  = 
6.0  +  or  F"  =  9°  32'. 


183.  In  case  no  frog  is  at  hand  of  the  angle  or  number  given 
by  eq.  (219)  or  (225),  we  may  select  one  as  nearly  like  it  as  pos- 
sible, and  locate  the  turnout  as  a  compound  curve,  pro- 
vided that  F"  is  less  than  2F.  Fig.  63. 


Fio.  63. 

Let  r"  =  C'a",  and  r  -  r'  =  QT  =  Cf 
Then  analogous  to  the  equations  of  §  180, 

(r»4-tt)=      -*- 
vers  ^" 

i« 
.-.  r"  = 


(226) 


(227) 


exsec  }!?" 

aF"  =  (r"  +  $g)  sin  ^^"  =  r"  tan  117"'        (228) 


TURNOUTS. 
The  length  of  the  switch,  by  eq.  (218),  is 


153 


AD= 


1  DK 


The  curvature  of  the  rail  between  the  frogs  F"  and  F  is 
F"CF  =  (F  -  iF"). 

Draw  the  chord  F"Fand.  the  perpendicular  F"L;  then  the 
angle  LFF"  =  F-  $(F  -  iF")  =  i(F  +  i^");  and  since 


LF-\g.  cot  i 


(329) 

(230) 

(281) 


Example.—  Let  F  =  6°  44'    . 

Eq.  (226)  iff        2.354 

iF" 

r"    569.616 
Eq.  (228)  iF" 

aF"      51.839 

Eq.  (229)  iff        2.354 

i(F+iF") 

F"F     22.645 

Eq.  (231)  i(F-  iF") 

2(r  +  iff)  1692.432 
r 


"'  =  10°  24' 


log  0.371806 
5°  12'    log  exs  7.616224 

2.755582 
5°  12'     log  tan  8.-959075 

1.714657 


When  ft"  >  .  707ft,  r  will  be 
equal  F,  (F"  being  given),  then 
also,  by  substituting  F'  for  ^i 


log  0.371806 
5°  58'     log  sin  9.016824 

O54982 
0°  46'     log  sin  8.126471 

3.228511 


less  than  r".    Should  F  '  not 
r'  and  J7.F7'  must  be  calculated 
eqs.  (230)  and  (231). 


184.  From  the  same  switch  in  a  straight  track  it  is  required 
to  lay  two  turnouts  on  the  same  side.  Fig.  64. 

If  we  assume  F'  =  F,  and  that  these  two  frogs  shall  be 
opposite  each  other,  we  calculate  all  the  distances  of  the  first 
turnout  for  the  angle  F  (or  number  ft)  by  §  180,  181,  whence 
we  have  the  radius  r  =  Ca. 


154 


MELD   ENGINEERING. 


Let  r'  =  C'a,  the  radius  of  the  centre  line  of  the  second 
turnout.  The  angle  AGF '=  P.,  and  since  F'  =  F,  the  angle 
CF'C'=  F,  and  the  triangle  CF'C1  is  isosceles,  and  C'F'  = 
C'C.  But  C'F1  =  C'A  = 


or 


(232) 
(233) 


C' 
FIG.  64. 


To  calculate  the  remaining  frog  at  F",  we  have  from  eq. 
(207) 


vers  F"  =  — 
or  from  eq.  (216) 

BF'  =  (?•' 
of"  =  2r'  sin  iJ 
and  since  AO'F'  =  2F, 


sin  F  = 
2r" 


of'  =  2r'  sin  F 


(234) 

(235) 
(236) 
(237) 

(238) 


The  length  of  switch  may  be  calculated  by  either  r  or  r', 
since  for  r',  which  is  about  ^r,  the  throw  of  switch  is  double, 
thus  giving  practically  identical  results. 

If  we  compare  the  values  of  F"  as  obtained  by  eqs.  (234) 
and  (219),  we  shall  find  them  almost  identical  for  given  values 


TURNOUTS. 


155 


of  jPand  g\  and  since  this  may  also  be  proved  analytically  by 
assuming  that  vers  $F"  =  i  vers  F\  which  is  very  nearly 
true  for  ordinary  values  of  F",  we  conclude  that  a  set  of  frogs 
(F  =  F' ,  and  F")  which  is  adapted  to  a  double  turnout  in 
opposite  directions  from  a  straight  line  (as  in  Fig.  62)  is  also 
adapted  to  a  double  turnout  on  one  side  (as  in  Fig.  64),  the 
curves  being  simple  curves  in  every  case.  But  this  being 
true,  the  set  is  also  adapted  to  a  double  turnout  in  opposite 
directions  from  any  curved  track  the  radius  of  which  is  not 
less  than  r  as  given  for  F,  since  any  such  case  is  intermediate 
between  the  two  cases  named.  When,  therefore,  a  certain 
frog,  F,  is  adopted  for  general  use  on  any  road,  another  frog 
should  also  be  adopted,  whose  angle,  F ",  is  determined  by 
eq.  (219),  or  whose  number  n  is  determined  by  eq.  (225). 
Thus,  if  F  =  6°  44',  or  n  =  8|,  then  F"  should  be  9°  32',  or 
n"  =  6. 

185.  In  case  no  frog  is  at  hand  of  the  angle  or  number  given 
by  eqs.  (234)  (235),  we  may  select  one  as  near  the  same  angle 
as  possible,  and,  calling  this  F",  calculate  the  distance  BF" 
and  the  radius  C"F"  (Fig.  65)  as  for  a  single  turnout;  §  180. 


Then  assuming  any  other  frog  F'.  whether  equal  to  .For  not, 
it  is  required  to  find  the  chord  F"F' ,  and  the  radius  C'F'  of 
the  arc  F"F'.  The  point  F'  may  fall  either  side 'of  the  radius 
CF,  according  to  the  values  given  to  F"  and  F'. 

a.  In  case  F'  falls  beyond  the  radius  CF,  we  will  assume 
first,  that  the  entire  rail  from  B  to  F'  is  laid  with  the  same 
radius  BC,  and  centre  C.  (This  investigation  also  applies  to 
the  case  when  F'  falls  between  B  and  the  line  CF.) 

In  the  diagram  (Fig.  65)  draw  CF\    We  then  have 


156  FIELD   ENGINEERING. 


and 

QF"  =  (r  —  ig)  exsec  BCF"  (240) 

In  the  triangle  F"CF', 

F"C  -  F'C  :  F"C  +  F'C  ::.  tan  $(F"F'C-  F'F"C) 

:  cot  F"CF' 
Now,  since  C'F'C  =  F',  and  BC"F"  =  F", 

.'.   F"F'C=  F"F'C'  +  F' 
and 

F'F"C=  F'F"O"  -  C"F"C=F"F'C'  -  (F"  -BCF") 
Letting  U-  C"F"C  =  (F"  -  BCF") 

and  subtracting,  we  have 

F"F'G-  F'F"O='F'  -\-  U 
Hence  the  above  proportion  may  be  written 

GF"  :  2BC  +  GF"  ::  tan  i(F'  +  U)  :  cot  %F"CF' 
whence 

cot  IF"CF'  =  2B°+p?F"  tan  l(F'  +U)         (241) 

(Since  BCF"  -f  F"CF'  =  BCF',  and  we  know  the  radius 
BC,  the  chord  or  arc  BF'  is  easily  obtained,  which  fixes  the 
position  of  the  frog  F';  and  the  problem  may  end  here, 
frequently,  in  practice.) 

Now  in  the  same  triangle  "'F'-'OF',  flic  half  sum  of  F"F'C 
and  F'F"C  is  90°  -  $F"CF';  while,  as  we  have  just  seen, 
the  half  difference  is  %(F'  -f-  £0;  and  by  subtracting  we  have 
the  less,  or 

F'F"C  =  90°  -  i(F'  +  17+  F"CF')          (242) 
F'  C  sin  F"CF' 


,_          Jg(7,  sin  ^GF^ 
°r  ^  F   ~  'r''r 


TUKKOUTS.  157; 

To  find  the  angle  F"C'F';  produce  the  line  F-'O'  in  the  dia- 
gram to  intersect  the  line  EG  at  K.  Then  the  two  triangles 
KC"C'  and  KCF'  have  the  angle  K  common,  and  the  sum  of 
the  other  angles  will  be  equal  ;  that  is, 

KC"C  +  'C"C  K  =  KCF'  -f  CF'K 

or  F"  +  F"C'F'  =  BCF'  +  F' 

and  since  BCF'  —  BCF"  +  F"CF' 

.-.  F"C'F'  =  F"CF'  +  F'  -  V"  (244) 

If  we  denote  the  radius  F'C'  by  r'  +  |^ 


<245> 


Example.—  Given:  the  three  frogs  ^  —  6°  43'  59",  ^"  = 
6°  01'  32",  and  ^"  =  8°  47'  51"  to  lay  a  double  turnout  on  one 
side  of  a  straight  track.  Fig.  65. 

By  Tab.  XI.  BF  =  80  036    r   =  680.306  AD  =  23.82 

BF"  =  61.204    r"  =  397.'  826 

Eq.  (239)  BF"      61.204  log  1.786779 

(r-ig)    677.952  "    2.831199 

BCF"  5°  09'  38"        log  tan  8^55580 


Eq.  (240)  BCF"  •<     5°  09' 38"    log  exsec  7.609587 

(r  -  ig)    677.952  log  2.831199 

GF"        2.760  "    0.440786 


Eq.  (241)  (2BC  +GF")  1358.664  '    3.133112 

2.692326 
tan  8.926968 

i(F"CF')  1°22'35"         "    cot  1.619294 


(£r=3°38'13ff)  2.692326 

i(F'  +  U)  4°  49'  52".5    log 


Eq.  (243)  F'CF-  2°  45'  10"         "   sin  8.681481 

r-tff    677.952  2.831199 


1.512680 

l(F'+  U+F"CF')  6°12'27".5      "   cos  9.997446 

F"F'      32.752  1.515234 

Eq.  (245)        $F"C'F'  2°  34'  14".  5      "    sin  8.651781 

730.219  2.863453; 

362.755 


158 


FIELD   EKGINEERIKG. 


b.   We  assume,  secondly,  that  the  middle  track  is  straight 
beyond  F,  and  tangent  to  the  curve  at  F.     Fig.  66. 

Then  whenever  the  value  of  F"  is  less  than  that  given  by 
eq.  (234),  the  arc  AF",  produced  with  the  same  radius  AC", 
will  intersect  the  straight  rail  HF'  at  some  point  F',  and  the 
frog  angles  F  and  F'  will  be  equal. 
-  ' 


FIG.  66. 

For  the  straight  rail  HF'  produced  backwards,  passes 
through  the  point  A,  making  an  angle  F  with  the  main 
track,  since  the  triangles  CBF  and  CHA  are  equal,  and  AH 
=  BF.  Now  any  circle,  tangent  to  the  main  rail  at  A, 
will  intersect  the  line  AH  in  some  point  F',  and  since  AF'  is 
the  chord  of  the  arc,  the  angle  at  F'  equals  the  angle  at  A, 
which  is  F.  Hence  F  —  F'  •  and  the  angle  AC"F'  =  2F, 

The  length  of  the  chord  AF  is 


AF'  =  2AC"  sin  F 
The  chord  F"F'  =  2F"C"  sin 


Hence, 


=  2AC"  sin 


F"F'  -  2(r"  -f- 


(246) 


(247) 


Example.—  Let  F'  -  F  -     6°  43'  59"  and  j^"  =  8°  47'  51* 


By  Table  XI.      r"  -  397.826 
Eq.(247)2(r"  +  ^)  =  800.360 

' 


2°  20'  03".5 


32.60 


log  2.903285 
log  sin  8.609915 

1.513200 


If  the  frog  F'  is  required  to  be  different  from  F,  then  the 
inside  curve  must  be  compounded  at  F",  giving  other  values 
to  the  length  and  radius  of  the  arc  F"F'. 


TURNOUTS. 


159 


c.  "We  assume,  thirdly,  that  the  curve  of  the  middle  track  is 
reversed  at  F.  Fig.  67. 

In  the  diagram,  let  Q  be  the  centre  of  the  reversed  portion, 
and  F'  the  proper  position  of  the  frog  F',  and  C'  the  centre 
of  the  required  arc  F  F'.  Then  Q  is  on  the  radial  line  CF, 
produced,  and  6"  is  on  the  radial  line  F"C"  produced.  Join 
FQ  and  F'Q,  and  produce  C"F"  to  intersect  these  lines  in  L 
and  M  respectively.  Also  join  F"Q,  and  denote  the  angle 
LF'Q  by  (/and  the  angle  F'QF"  by  Q. 


FIG.  67. 

In  the  triangle  FF"Q  we  know  F"F=  BF-  BF",  and  the 
side  FQ  is  given;  and  the  included  angle  F"FQ  =  90°  -f-  F. 
Hence  we  may  calculate  (Tab.  II.  25)  the  angle  F"QFa.nd  the 
side  F"Q. 

The  triangle  CC"L  gives  the  angle  at  L  =  F"  —  F;  and  the 
triangle  F"LQ  gives  LF  'Q  =  L  —  F"QF 

.-.    U=  F"  -  F-  F"QF  (248) 

In  the  triangle  F'QF"  we  have 
F'Q  -  F"Q  :  F'Q  -f  F"Q  ::  tan  $(F'F"Q  -  F"F'Q) 


But  F'F"Q  =  F'F"L  +  £7  and  F"F'Q  -  F"F'N-  F',  and 
since  F"F'N  =  F'F"L,  we  have  by  subtraction, 

F'F'-Q-  F"F'Q  =  U-\-F' 
Hence         cot  4  Q  =  *j&Jr*£&  ten  I  (U+F')        (249) 


160  FIELD    ENGINEERING. 

(Now  the  angle  FQF'  =  Q  -  F"QF,  and  is  subtended  by  the 
chord  JIF',  which  is  therefore  easily  found,  and  serves  to 
locate  the  frog  F',  and  frequently  this  is  all  that  will  be 
required.) 

In  the  triangle  F"QF',  the  half  sum  of  QF"F'  and  QF'F" 
is  90°  .—  iQ,  while,  as  we  have  just  seen,  the  half  difference  is 
F');  hence  by  adding,  we  have  the  greater,  or 

'  -  Q) 

•  (250) 


The  triangle  C'F'M  gives  F"G'.F'  =  P'  —  M,  while  the 
triangle  F"MQ  gives  M=  U  -\-  Q;  hence  F".C'F'  =  F'  - 
Q);  and  denoting  the  radius  G  'F'  by  r'  -j-  $g, 


Example.—  Lei  F  =  F'  =  6°  43'  59",  F"  =  8°  47'  51",  and 
FQ  =  953.012.  Then  by  Tab.  XL,  BF  '=  80.036  and  BF  "  = 
61.204;  hence  ^"^  =  18.832;  and  the  included  angle  is 
96°  43'  59". 

Solving  the  triangle  FF"Q  we  find  F"QF  =  1°  07f  18", 
FF"Q  =  82°  08'  43",  and  F"Q  =  955.402.  Now  FQ  = 
^4-^  =  957.720. 

(249)    F'Q+F"Q    1913.122  log  3.281743 

F'Q-F"Q         2.318  "  0.365113 

(CTO°56'34")  "   ^916630 

F')  3°  50'  16".  5  log  tan  8.826231 

1°  02'  08".  4     "   cot  1.742861 


(250)  0  2°  04'  16".8     "  sin  8.558033 
l(U+  F'-  Q)  2°  48'  08".l     "  cos  9.999480 

8.558553 

F'Q  957.720                                   log  2.981239 

F"F'  34.633                                         1.539892 

(251)  #F'  -U-Q)  1°  51'  34".l   log  sin  8.511191 

2(r'  +  4#)  1068.32                                           3~J028701 

r'  53181 


TURNOUTS.  161 

186.  Given:  a  main  track,  curved,  and  a  frog-angle  F,  to 
locate  a  turnout  on  tJie  inside  of  the  curve.  Fig.  68. 

Let  R  =  Oa     =  radius  of  main  track. 
"    r  =  Ca     =  radius  of  turnout. 
"    F  =  CFO  =  the  frog  angle. 

In  the  diagram  draw  the  chord  JLFand  produce  it  to  inter- 
sect the  outer  rail  at  G;  and  draw  FO  and  GO.  Since  the 
chords  AF  and  AG  coincide,  and  the  radii  AC  and  AO 


coincide,  the  chords  subtend  equal  angles  at  C  and  0  respec- 
tively, and  GO  is  parallel  to  FG.  .(See  §137.)  Hence,  FOG 
=  CFO  =  F.  Let  6  =  the  angle  FOA. 

In  the  triangle  FOA,  0  =  GFO  -  FAO  =  GFO  -  FGO; 
and  in  the  triangle  GFO,  GO+  FO  :  GO  —  FO  ::  tan  l^GFO 
+  FGO)  :  tan  i(GFO  -  FGO),  or  ZR  :  g  ::  cot  $F  :  tan  $0 


.-.  tan  |9  =  --  cot^=.-  (252) 

In  the  triangle  CFO, 

(y+ig)  =  (B-^)sinsj,%    -      <253> 

In  the  triangle  EOF, 

BF  =  2(R  -  ig)  sin  |0  (254) 

In  the  triangle  aCf, 

af  =  2r  sin  #F+  6)  (255) 


162 


FIELD   ENGINEEKII^G. 


The  length  of  switch  AD,  for  a  given  throw  DK,  may  be 
found  thus:  from  Table  IV.  take  the  tangent  offsets,  t  and  t', 
corresponding  to  R  and  r  respectively,  and  assuming  that  the 
offsets  may  vary  as  the  squares  of  their  distances  from  the 
tangent  point,  we  have 

t-t'   :  DK::  (1/)0)2  :  AD* 


AD  = 


V 


*.-  a 


(256) 


This  result  is  practically  the  same  as  that  found  for  length 
of  switch  in  a  turnout  from  a  straight  line  with  the  same  frog, 
when  R  is  large. 

Example.— Let  R  =  1432.69  and  F  =  6°  43'  59". 


Eq.  (252) 


2.354 


R  (Tab.  IV.) 


3°  21'  59". 5 


log  0.371806 
log  cot  1.230440 

log  1.602246 
"  3.156151 


1°  35'  59". 8    log  tan  8.446095 


Eq.  (254)          0 

F+e 


3°  11'  59". 6 
9°  55'  58".6 


(254) 


(255) 


R-ig  1430.336 

r  +  iff  462.856 

r  460.502 
2 

R  —  4<7)  1430.336 


sin  8. 746786 
"  9.236778 

9^510008 
3.155438 


BF 

2r 


79.872 
921.004 


1°  35'  59".8 


4°  57'  59".3 


of       79.734 


2.665446 

log  0.301030 

"  3.155438 

log  sin  8.445924 

log:L902392 

"  2.964262 

log  sin  8.937381 

log  1.901643 


The  values  of  BF  and  af  are  found  to  be  so  nearly  identical 
in  this  case  with  those  determined  in  case  of  a  turnout  from  a 
straight  line,  that  the  values  given  in  Table  XI  may  be  used 
at  once  for  ordinary  values  of  R;  and  the  degree  of  curve  of  the 
turnout  in  this  problem  is  approximately  the  sum  of  the  degree 
of  curve  of  the  main  track  and  the  degree  of  curve  given  in 
Table  XI.  opposite  F.  Thus,  in  the  example  4°  -f-  8°  26'  = 
12°  26'  .-.  r  =  461. 7  nearly. 


TURNOUTS. 


163 


187.  Given:  a  main  track,  curved,  and  a  frog-angle  F,  to 
locate  a  turnout  on  the  outside  of  tJie  curve.  Fig.  69. 

In  the  diagram  draw  the  chord  AF,  and  produce  it  to  meet 
the  inner  rail  at  G;  and  draw  FO  and  GO.  The  triangle? 
CAF  and  OA G  are  both  isosceles,  and  have  the  angles  at  A 
equal;  hence  they  are  similar,  and  FCA  =  AOG.  Hence 
FOG  =  HFO  =  F.  Let  R  =  Oa,  r  =  Ca,  and  0  =  FOA, 


FIG.  69. 

In  the  triangle  FOA,  0  =  OAG  -  AFO  =  FGO  -  GFO; 
and  in  the  triangle  FOG;   FO  -f-  GO  :   FO  —  GO  ::  tan 
i(FOO  +  GFO)  :  tan  $(FGO  -  GFO),  or  2R   :   g  ::  cot  $F 
:  taniQ 


which  is  identical  with  (252). 
In  the  triangle  CFO 


In  the  triangle  BOF, 

J5^=2CK  +  ^)sin40 
In  the  triangle  a  Cf, 

af=2r.  sm^(F-G) 

For  a  given  throw,  the  length  of  switch  will  be 


AD  =  j/10000 


(257) 

(258) 

(259) 
(260) 

(261) 


164  FIELD   ENGINEERING. 

in  which  t  and  t'  are  the  tangent  offsets  (Tab.  IV.)  corre- 
sponding to  R  and  r. 

In  this  problem,  as  in  the  preceding,  we  may,  for  ordinary 
values  of  R,  assume  the  values  for  BF and  a/given  in  Tab.  XI. 
The  degree  of  curve  of  this  turnout  is,  approximately,  d  —  D, 
taking  d  from  Tab.  XI.  and  D  from  Tab.  IV.  corresponding 
to  R.  Should  D  =  d,  this  turnout  becomes  a  straight  line; 


FIG.  70. 

and  when  D  >  d,  or  when  R  is  less  than  r  given  in  Tab.  XI., 
the  centre  falls  on  the  same  side  as  0.  Fig.  70.  In  this  case, 
using  the  same  notation,  £0  is  given  by  eq.  (257). 


Eq.  (259)  BF  =  2(R  +  $g)  sin  £0 

of  =  2r  sin  i(B  -  F)  .  (263) 

188.  A  tongue-switch  is  a  short,  stiff  switch  which, 
when  moved,  revolves  at  the  heel  as  on  a  pivot.     When  it  is 
thrown  over  to  the  turnout  track,  it  makes  an  abrupt  angle 
with  the  main  track,  called  the  switch  angle;  but  in  this  posi- 
tion it  should  be  tangent  to  the  turnout  curve.     The  use  of 
this  switch  is  generally  confined  to  yards  and  warehouses, 
where  but  little  space  can  be  afforded,  and  where  the  motion 
of  the  cars  is  always  slow. 

189.  Given:  a  straight  track,  a  frog-angle  F,  and  the  length 
and   throw   of  a   tongue-switch,  to   locate   the  turnout. 
Fig.  71. 


TURNOUTS.  165 

Eet  AD  be  the  length,  and  DK  the  throw  of  switch,  and  let 
S  denote  the  switch-angle  DAK. 


T)  Jf 

Then          sin  8  =  ~~  or  8°  =  57°.3      ~-  (264) 

(Compare  §86.) 

Let  C  be  the  centre  of  the  required  turnout,  and  in  the  dia- 
gram draw  CK  and  GF\  also  draw  DG  perpendicular  to  the 
.straight  track.  Then  DGF  =  F;  and  in  the  triangle  KGC, 
KCF  =  KGF  -GKC,  and  since  CKA  is  a  right-angle,  GKG 
-  8  .-.  KCF=  F-  8. 

Draw  the  chord  KF,  and  since  the  triangle  KCF  is  isosceles, 
the  angle  CFK  =  90°  -  \(F  —  8).  Now,  CFI  =  90°  -  F\ 
hence  by  subtraction,  KFI  =  i(F  +  8). 


FIG.  71. 


If  g  denote  the  gauge,  we  know  KI  =  g  —  DK-  and  in  the 
right-angled  triangle  KIF,  we  have 

IF=  KI  .  cot  $(F+  8)  (265) 


(367) 


These  equations  are  analogous  to  eqs.  (229)  (230)  (231). 

19O.  Given:  a  double  turnout  with  tongue- 
switch,  from  a  straight  track;  to  find  the  angle,  F",  of  the 
middle  frog. 

Assuming  F'  =  F  calculate  (r  -f  |#)  by  the  last  equations. 
Since  the  rails  of  the  turnouts  intersect  on  the  centre  line  .of 


166 


FIELD 


the  straight  track,  as  in  Fig  63;  if  we  substitute  the  value  of 
F"  F',  eq.  (229)  in  eq.  (231),  we  have 

__lff 

F")sini(F-  iF") 

iff 


2  sin 
and  by  Trig.  Table  II. 


cos^F"  —  cosF 


whence 


r  +  iff) 


(268) 


If  the  angle  of  the  middle  frog  to  be  used  does  not  agree 
with  F"  found  by  the  last  equation,  the  turnout  will  be  com- 
pounded at  F". 

191.  Given :  a  straight  track,  tfie  frog-angles  Ff   F'  and  F", 
and  the  switch  angle  S,  to  locate  a  double  turnout. 

Fig.  72. 


FIG.  72. 

Assuming  that  F"  shall  be  placed  on  the  centre  line  of 
the  straight  track,  let  h  be  a  point  on  the  centre  line  at  the 
point  of  switch.  Then  JiK  =  ig  —  DK;  and  since  the  angle 
F"  is  bisected  by  the  centre  line  the  necessary  formulae  in  this 
case  are  obtained  from  §189  by  simply  replacing  ^byl^" 
and  KI  by  hK;  and  in  the  first  members  ZZ^by  hF"  and  r  by 
r\  This  is  obvious  by  the  similarity  of  the  figures. 


TURNOUTS. 

Hence  hF"  =  hK. 

'  •""     KF"  = 


hK 


sin 


-  S) 


167 
(269) 
(270) 

(271) 


The  location  of  the  remaining  frogs  is  a  problem  already 
discussed,  §  183,  eq.  (229),  etc. 

192.  Given:  a  straight  track,  the  frog  angles  F,  F',  F",  and 
the  switch  angle  S,  to  locate  a  double  turnout  on  one 
side.  Fig.  73. 


FIG.  73. 

The  frog  Fis  located  by  §  189;  but  for  the  frog  F"  we  have 
evidently  a  double  throw;  hence  eqs.  (265)  (266)  (267)  become 


IF"  =  (g  -  2DK)  cot 


28) 


sn 


sin  i(F"  -  28) 


(272) 
(273) 

(274) 


To  locate  the  remaining  frog  F' :  when  F'  falls  beyond  the 
line  CF,  there  are  three  cases. 

a.  The  middle  track  reversed  beyond  F. 

We  find  the  distance  F"Fby  aubtracting  IF",  eq.  (272)  from 
IF,  eq.  (265) :  after  which  the  solution  is  identical  with  that 
given  §  185,  C.,  Fig.  67. 


168  FIELD 


b.  The  middle  track  compounded  at  F. 

Let  Q  be  the  centre  of  the  curve  beyond  F,  and  also  let  Q  = 
the  angle  F'QF";  and  let  U  =  the  angle  C"F"Q. 

Then  by  a  course  of  reasoning  analogous  to  that  of  case  a, 
we  derive 

...         U=F"-F+F"QF  (275) 

cot  IQ  =  y'.Q  +  y.Q  tan  i(U+  F')          (276) 

Now  since  the  radius  F'Q  is  given,  and  the  angle  FQF1  = 
Q  —  FQF",  we  readily  determine  the  distance  HF',  and  so 
locate  the  frog  F'. 

In  the  triangle  F"QF',  the  half  sum  of  QF"F'  and  QF'F" 
is  90°  -  $Q,  while  the  half  difference  is  $(U  +  F');  hence  by 
subtraction  we  have  the  less,  or 

F'F"Q  =  90°  -  i(  U+  F'  +  Q) 
Hence  P'F'  =  FQ—-^^^  (277) 

Join  C'Q,  and  the  quadrilateral  C'QF'F"  gives 
F'  +  Q  =  U+F"C'F' 

hence  F"O'F'  =  F'  -  U+  Q',  and  denoting  the  radius  C'F 
by  r'  -j-  $g,  we  have 


Cor.  Since  the  centre  Q  is  assumed  at  pleasure,  it  may  be 
made  to  coincide  with  the  centre  C,  and  then  the  compound 
curve  becomes  a  simple  curve.  Then  also,  the  above  formulae 
will  apply  when  F'  is  such  that  the  frog  will  come  on  the  arc 
Iff.  But  as  FQF"  will  be  greater  than  Q,  the  difference 
FQF'  will  be  negative,  indicating  that  the  distance  HF'  is  to 
be  laid  off  backwards  from  H. 

c.  The  middle  track  straight  beyond  F,  and  tan- 
gent to  the  curve  at  F.  Fig.  74. 

Let  F'  be  the  required  position  of  the  frog  F'.  A  tangent 
to  the  curve  at  F'  makes  an  angle  (F'  -f-  F)  with  the  main 
track,  and  a  tangent  at  F"  makes  an  angle  of  F"  with  the 
same;  hence  the  angle  they  make  with  each  other  is 


TURNOUTS. 


169 


'—  F"),  and  this  is  the  curvature  of  the  arc  F"F't 
and  equals  the  angle  F"C'F'. 

Produce  the  straight  line  F'H  backwards  to  G,  and  draw 
F"G  perpendicular  to  it.     Then  F"G  =  FH—  F"F.  sin  F,  or 

(279) 


FIG.  74. 

In  the  right-angled  triangle  F'GF",  the  angle  F"F'G  = 
F'  -  %(F'  +  F-  F")  =  i(F'  +  F"  -  F). 


F"F'  =  -? 


F"G 


sin  i(F'  +  F"  -  F) 
and  GF'  =  F"F'  .  cos  \(F'  +F'.-F)  (281) 

Observe  that  GF'  cannot  be  less  than  GH=  F"F.  cos  F. 


193.  Given:  a  turnout  with  a  frog  angle  F,  and  the  perpen* 
dicular  distance  \>  between  the  centre  lines  of  the  main  and  side 


FIG.  75. 

tracks ;  to  find  the  radius  r  of  the  curve  connecting  the 
turnout  with  the  side  track.    Fig.  75. 


170  FIELD   ENGINEERING. 

Let  the  reversing  point  be  taken  at  F,  and  let  Q  on  CF  pro- 
duced be  the  centre  of  the  required  curve,  and  draw  Q,M  per- 
pendicular to  the  main  track.  Then  QM=  QF=  r  —  ^g;  the 
point  M  is  the  point  of  tangent,  and  the  angle  FQM  =  F. 

Now  N  being  the  intersection  of  the  rail  .B^with  the  radius 
QM,  we  have  MN=  QFvers  F,  but  MN  =  p  —  g;  hence 


The  distance  FN  is  evidently 

FN=(r-ig)smF  (283) 

and  the  chord  to  the  centre  line  is 

/ra  =  2r  sin  \F  (284) 

Should  the  distance  FN  consume  too  much  of  the  track,  it  may 
be  lessened  by  introducing  a  short  tangent  at  F,  denoted  by  k; 
then  by  eq.  (48)  the  radius  will  be  shortened  by  an  amount 
equal  to  k  .  cot  $F,  and  the  distance  FN  will  be  shortened  by  k. 

Since  the  tangent  k  reduces  the  length  of  the  tangent  offset 
of  the  entire  curve  by  k  .  sin  F,  we  have  for  the  new  radius  r' 


When  r'  is  fixed  by  a  limit,  we  obtain  k  by  resolving  eq.  (285) 

p-.g-(r'  -^  VGT3f> 
TO*" 

In  case  the  main  track  is  but  slightly  curved,  we  may  at  first 
assume  it  to  be  straight,  and  find  r  as  above,  eq.  (282),  and 
the  degree  of  curve  corresponding  to  r;  but  this  degree  of 
curve  must  then  be  increased  or  diminished  by  the  degree  of 
curve  of  the  main  track,  according  as  the  track  is  concave  or 
convex  toward  Q. 

194.  Given :  the  perpendicular  distance  p  between  the  centre 
lines  of  a  curved  main  track  and  a  parallel  side  track,  and  the 
frog  angle  Fofa  turnout;  to  find  the  radius  r  of  the  connecting 
curve,  and  the  length  FIST,  or  fin,  of  the  curve.  Fig.  76. 


TURNOUTS. 


171 


Let  FN  be  the  rail  of  the  main  track,  and  GM  the  rail  of 
the  siding,  adjacent  to  each  other;  let  0  be  the  centre  of  the 
main  track,  and  Q  the  centre  of  the  connecting  curve.  Then 
the  connecting  curve  will  terminate  at  m,  on  the  line  OQ  pro- 
duced. 

In  the  diagram  draw  MF,  and  produce  it  to  intersect  the 
rail  MG  at  G,  and  join  GO,  FO,  and  FQ. 

Let  R  =  radius  of  centre  line  of  the  main  track;  r  =  radius 
of  centre  line  of  the  connecting  curve;  and  0  =  the  angle 
FOM. 

Case  a. — The  siding  outside  the  main  track.    Fig.  76. 


FIG.  76. 


By  similarity  of  the  triangles  GOM  and  FQM,  GO  is  paral- 
lel to  FQ,  and  the  angle  GOF  =  F;  and  by  a  process  similar 
to  that  of  §  186,  we  have 


(287) 
(288) 


sinB 


sin  (F-\-  0) 


fm  =  2r  .  sin  #F+  0)  (290) 

Case  b. — The  siding  inside  the  main  track.     Fig.  77. 
By  a  process  entirely  similar  to  §  187,  we  have 


cot  & 


(291) 


172 


FIELD   ENGINEERING. 


r  -  to  =  (R  -  i 
FN  =  2(R  - 


sin  0 


sin  (F—G) 
)  sin  *0 


fm  -  2r  sin  \(F  —  0) 


(292) 
(293) 
(294) 


When  0  =  .Fin  the  last  equations,  sin  (F  —  6)  =  0,  and  r 
is  infinite,  and  the  curve  FM  becomes  a  straight  line. 


FIG.  77. 


When  0  >  F,  sin  (JF—  6)  is  negative,  and  the  centre  Q  falls 
on  the  same  side  of  the  track  as  0,  and  we*  have 


fm  =  2r  .  sin  |(6  —  F) 
Equations  (291)  and  (293)  remain  unchanged. 


(295) 
(296) 


195.  To  locate  a  crossing  between  parallel  tracks. 
Fig.  78. 

When  a  turnout  from  one  track  enters  a  parallel  track  by 
means  of  another  frog  and  switch,  the  whole  is  called  a  cross- 
ing. The  frogs  are  alike,  and  the  calculation  for  one  end  of 
the  crossing  answers  for  the  other.  §§  180,  181.  We  have 
only  to  find  the  length  of  track  between  the  two  frogs. 

In  the  diagram  let  AF  be  one  turnout,  and  A'F'  the  other, 
connected  by  the  straight  track  F'G.  It  is  required  to  deter- 
mine the  length  F'G,  or  the  distance  FN  measured  on  the 
mam  track  from  F  to  a  perpendicular  through  F'.  Produc- 
ing the  line  F'G  to  intersect  the  rail  NFat  H,  we  have  two 


TURNOUTS. 


173 


right-angled  triangles  GFH  and  F'NII,  having  the  common 
angle  at  H  —  F.  Let  p  =  the  perpendicular  distance  between 
centre  lines  of  main  tracks,  and  g  —  gauge.  Then  GF  —  g, 
=  (p-ff.) 


F'G  =  F'H  -  OH  =     -       -  GFcot  F 


(297) 


So 


FN  =  Nil  -  FH  =  (p-g)  cot  F  - 


sinF 


(298) 


W7ien  the  main  tracks  are  curved  the  distance  F'G  may  be 
calculated  by  the  same  formula  (297)  which  gives  a  value  only 
a  fraction  too  small,  but  in  laying  the  track  the  rail  F  G  must 
be  curved  to  a  radius  which  is  to  R  of  the  main  track  as 
F'G  :  NF. 


When  p  is  large,  or  the  tracks  are  very  wide  apart,  it 
will  effect  some  saving  of  room  to  lay  Hie  crossing  in  the  form 
of  a  reversed  curve ;  and  the  frogs  being  alike,  the  two 
arcs  will  be  equal,  and  the  point  of  reversed  curve  P  will  be 
midway  between  ^aud  F'.  Fig.  79. 


FIG.  79. 


In  the  diagram  we  have  aPa'  the  centre  line  'of  the  cross- 
ing, and  PL  the  centre  line  between  .tracks;  aL  =  $p,  and 
aC  —  a'C'  =  r.  The  radius  r  having  been  found  by  §  180  or 
§  181,  we  have 


and 


vers  aCP  —      - 
PL  —  r  sin  aCP 


(299) 
(300) 


174 


FIELD   ENGINEERING. 


The  distance  between  frogs,  FN,  measured  on  the  main  track 
is  evidently 

FN  —  ^(PL  -  BF)  (301) 


in  which  BFis  determined  by  eqs.  (209),  (213),  or  by  Tab.  XL 

197.   To  lay  a   crossing  in  the  form  of  a  reversed 
curve,  when  the  parallel  tracks  are  on  a  curve.    Fig.  80. 


Let  0  be  the  centre  of  the  main  curve,  G  and  C'  the  centres 
of  the  reversed  curve. 

Then  in  the  triangle  GOG1  we  know  all  three  sides;  for  CO 
—  R  +  ?•;  GG'  =  r  +  r't  and  C'O  =  R  -j-  p  —  r' ;  and  the  half 
sum  of  the  three  sides  is  s  —  R  -\-  r  -J-  %p. 

Denoting  the  angle  COG'  by  <p,  we  have  (Trig.  Tab.  II.  31) 


vers  a>  - 
vers- 


P  (r  ~  r>  ~ 


(302) 


The  angle  <p  determines  the  length  of  the  arc  BN  described 
with  the  radius  (R  +  £#)  and  so  fixes  the  position  of  the  point 
A  from  A. 
By  a  formula  similar  to  the  above, 

^1I|S^|>=S&^  (303) 


TURNOUTS.  175 

The  angle  C'CO  determines  the  length  of  the  arc  aP 
described  with  the  radius  r;  the  angle  ((p  -f-  C'CO)  =  CC'A 
determines  the  length  of  the  arc  Pa',  and  P  is  the  point  of 
reversed  curve. 

In  this  problem  R  is  known,  r  is  found  by  §  187,  and  r'  is 
found  by  §  186,  only  observing  that  in  this  case  the  value  of 
E  must  be  increased  by  "p.  The  frog  angles  7^  and  F'  may  be 
equal  or  otherwise,  only  taking  care  that  the  point  P  shall  be 
included  between  the  radii  G'F'  and  GF. 

The  angle  FOG  =  0  is  given  by  eq.  (257),  and  the  angle 
F'OC'  —  6'  is  given  by  eq.  (252)  (in  which  the  value  of  R  is 
to  be  increased  by  p);  hence  the  angle  FOF*  =  <p  —  (0  -|-  6'), 
which  determines  the  distance  between  the  frogs,  measured  on 
the  main  track. 

198.  To  find  the  middle  ordinate  m,  for   1  sta- 
tion,  or  100  feet,    on  any  curve,  m  terms  of  the  degree  of 
curve  D. 

Referring  to  Fig.  4  we  have  in  the  right  triangle  AGH 

OH  =  OA  .  tan  GAH 

But  GA  =  ±AB  =  $Ct  and  (Tab.  I.  18)  GAH  =  ±AOB  =  i  A ; 
hence 

Jf  = -IC.  tan±A  (304) 

a  general  expression  for  the  middle  ordinate  of  any  chord. 

If  in  this  equation  we  make  C  —  100,  A  becomes  D;  and 
denoting  the  corresponding  value  of  Jbf  by  m,  we  have 

m  =  ilOO  tan  £D  (305) 

whence  the  rule,  Multiply  the  nat  tangent  of  ±  the  degree  of 
curve  by  100  and  divide  by  2.  Thus  the  values  of  m  in  the  5th 
column  of  Tab.  IV.  have  been  calculated 

199.  To  find  the  middle  ordinate  for  any  chord  in 
terms  of  the  chord  and  radius 

Referring  to  Fig.  4  we  have 


GH  ~  OE  -  OG  =  OE  -  VAO*  -  GA* 


or  M**R-       £8__    I  (306) 


176  FIELD    ENGINEEBItfG. 

When  C  —  100  we  have  for  the  middle  ordinate  of  one 
station 


m  =  R-  V^2-2500  (307) 

For  any  subchord  c,  less  than  100,  we  have  for  the  middle 
ordinate, 


or  _  I-          (308) 


c4 
By  adding  ^—  -  to  the  quantity  under  the  radical  in  eq.  (308) 


it  becomes  a  perfect  square,  giving 

m,  =  ~  nearly,  (309) 

which  is  a  very  useful  formula,  although  approximate.  The 
error  in  mi  does  not  exceed  .002  for  any  subchord  c  when  the 
radius  is  greater  than  800.  On  a  20°  curve  the  error  will  be 
.002  for  a  chord  of  50  feet;  and  on  a  40°  curve  the  error  in  rrn 
will  be  only  .003  fora  chord  of  33  feet.  Equation  (309)  is 
therefore  practically  correct  in  all  cases  for  finding  the  middle 
ordlnates  of  rails.  Table  XII.  is  calculated  by  eq.  (308). 


2OO.  Curving  Rails.  Before  any  rail  is  spiked  to  its; 
place  in  a  curve,  it  must  be  evenly  bent  from  end  to  end,  so 
that  it  will  assume  the  proper  curvature  when  lying  free. 
The  bending  may  be  done  by  using  sledges,  but  is  best  accom- 
plished, especially  for  turnouts  and  other  sharp  curves,  by 
using  a  bending  machine  made  especially  for  this  purpose. 

The  proper  curvature  of  a  rail  is  tested  by  measuring  its 
middle  ordinate  from  a  small  cord  stretched  from  end  to 
end  and  touching  the  side  of  the  rail-head.  The  cord  should 
also  be  stretched  from  the  middle  point  of  the  rail  to  either 
end,  and  the  middle  ordinate  of  each  half  length  measured, 
to  test  the  uniformity  of  curvature. 

From  the  last  equation  it  appears  that,  with  a  given  radius, 
the  middle  ordinate  varies  nearly  as  the  square  of  the  chord. 


TURNOUTS.  177 

We  may  therefore  find  the  middle  ordinate  of  a  rail  whose 
length  is  c  by  the  proportion 

(100)2   :  c2  ::  m  :  m^ 

mi  =  -      nearly'  (310) 


in  which  m  is  obtained  from  Tab.  IV.,  col.  5,  for  the  given 
radius  or  degree  of  curve. 

Example.  —  What  is  the  middle  ordinate  of  a  30  ft.  rail 
when  curved  for  a  20°  curve? 


"When  a  long  rail  is  bent  for  a  sharp  curve,  observe  that  c  is 
the  length  of  the  chord  of  the  rail  —  not  of  the  rail  itself. 

For  the  chord  of  half  a  rail  the  middle  ordinate  is  one-fourth 
the  middle  ordinate  of  the  whole  rail.  Thus,  in  the  above  ex- 
ample it  would  be  .099  or  1T\  inches. 

Instead  of  using  the  chord  of  the  whole  rail,  it  may  be  more 
convenient  to  assume  a  chord  shorter  than  the  rail,  especially 
when  the  chord  is  not  an  exact  number  of  feet,  knotting  the 
string  to  the  length  assumed,  and  applying  it  to  different  por- 
tions of  the  rail  successively, 

20  1.  Elevation  of  the  outer  rail  on  curves. 

When  a  car  passes  around  a  curve,  a  centrifugal  force  is 
developed  which  presses  the  flanges  of  the  wheels  against  the 
outer  rail.  This  force  acts  horizontally,  and  varies  as  the 
square  of  the  velocity,  and  inversely  as  the  radius  of  the 
curve.  Denoting  the  centrifugal  force  by/,  we  have  from  the 

theory  of  mechanics  /  =  -^T^OO^D*  m  which  w  =  weight  of 
O4.  lob  H  -•"»..• 

loaded  car  in  pounds,  v  =  velocity  in  feet  per  second,  and 
R  =  radius  of  curve  in  feet. 

In  Fig.  81,  let  ab  represent  a  level  line  at  right  angles  to  the 
track,  let  a  and  c  be  the  tops  of  rails  on  a  curve,  let  be  —  e  =="' 
elevation  of  outer  rail  c,  and  let  the  point  d  be  the  centre  of 
gravity  of  the  car.     The  force  /  acts  in  the  direction  ab,  and 
if/'  =  the  component  of  /in  the  direction  ac,  then 

/'  :/  ::  ab  :  ac. 


178  FIELD   ENGINEERING. 

The  weight  w,  resting  on  the  inclined  plane  ac,  developes  a 
component  in  the  direction  ca,  and  denoting  this  by  w',  we 
have  by  similar  triangles, 

w'  :  w  ::  be  :  ac. 


FIG.  81. 


Since  equilibrium  requires  that  w'  shall  equal/',  we  have  after 

dividing  one  proportion  by  the  other  .-*—  =  —  -,  or  /=  -W^-. 

w         ao  ao 

Equating  this  value  of  /with  that  given  above  we  find, 


32.166.ff 


But  ah  —  V  ac*  —  e*,  and  ac  =  distance  between  rail  centres  = 

ROQA 

gauge  +  one  rail  head  =  g  -f-  0.  188.     Also  «  =  -g|S  V,  if  V  de- 


note  the  velocity  in  miles  per  hour.     Making  these  substitu- 

tions and  reducing,  we  have 

y-2 

.06688-- 
e  =  (g  +  .188)  --      _g  ______  (311) 


/         /  y* 

j/  1+  f.06688-jg) 


By  this  formula  Table  XIII.  is  calculated  for  the  standard 
gauge  g  =  4'  8|",  =  4.708. 

An  approximate  formula  may  be  obtained  by  assuming  that 
ab  —  g  for  practicable  values  of  e.     Substituting  this  in  the 

5280 
first  value  of  e  given  above,  and  replacing  v  by  -HTT^  K2  we 

have 

(approx.)  e  =  .06688-  (312) 

which  is  the  formula  generally  employed. 


TURNOUTS.  179 

In  laying  a  new  track,  the  transverse  inclination  is  first 
given  to  the  ballast  by  grade  pegs  driven  either  side  of  the 
centre  line  at  a  distance  of  (g  -f-  .188)  each  side  of  the  centre; 
the  outside  peg  being  set  higher,  and  the  inside  peg  lower 
than  the  grade  of  ballast  on  the  centre  line,  by  the  proper 
elevation  selected  from  Table  XIII.  But  in  re  surfacing  an 
old  track,  the  inner  rail  is  taken  as  grade  and  the  outer  rail  is 
raised  the  necessary  amount. 

2O2.  The  proper  elevation  may  be  found  mechan- 
ically by  the  following  method: 

To  find,  on  a  curved  track,  the  length  of  a  cJwrd  whose  middle 
ordinate  shall  eqpal  the  proper  elevation  of  the  outer  rail  for  any 
velocity  V.  in  miles  per  hour. 

By  the  conditions  of  the  problem,  we  have  m\  in  eq.  (309) 
equal  to  e  in  eq.  (312),  or 

c^    _  gV*  .  06688 
8£~  R 


c  =  .73144  Vg  (313) 

Wheug  =  4.708, 

c  -  1.587  F  (314) 

Lay  off  the  chord,  c,  upon  the  rail  of  the  track,  stretch  a 
piece  of  twine  between  the  points  so  found,  and  measure  the 
middle  ordinate;  it  will  equal  the  proper  elevation. 

2O3.  The  velocity  assumed  in  the  preceding  formulae 
should  be  that  of  the  fastest  regular  trains  which  will  pass 
over  the  curve  in  question,  since  the  flanges  would  be  forced 
against  the  outer  rail  were  there  no  centrifugal  force  devel- 
oped, by  reason  of  the  wheels  being  rigidly  attached  to  the 
axles,  and  the  axles  being  parallel. 

The  rails  on  tangents  should  be  level  transversely,  except 
near  curves,  where  for  50  or  100  feet  from  the  curve  one  rail 
is  gradually  raised,  so  that  at  the  P.  C.  or  P.  T.  it  may  have 
the  full  elevation  due  to  the  curve.  At  a  P.  C.  C.  th'e  elevation 
should  be  an  average  of  the  elevations  due  to  the  two  arcs. 
Owing  to  the  difficulty  of  properly  adjusting  the  elevation  of 
rail,  it  is  objectionable  to  have  arcs  of  very  dissimilar  radii 
join  each  other;  and  the  objection  is  much  greater  in  the  case 
of  reversed  curves  unless  separated  by  a  short  tangent.  See 
%S2. 


180  FIELD  ENGINEERING. 

On  the  other  hand,  a  short  tangent  between  arcs  which 
curve  in  the  same  direction  should  be  avoided,  since  it  makes 
a  "flat  place"  both  in  line  and  levels,  at  once  unsightly  and 
injurious  to  the  rolling  stock. 

In  the  case  of  turnouts,  however,  no  elevation  of  rail  is  pos 
sible  (except  when  both  tracks  curve  in  the  same  direction); 
hence  reversed  curves  are  allowable,  the  speed  of  trains  being 
usually  quite  low  also. 

2O4.  The  coning  of  the  wheels,  by  which  the 
wheel  on  the  outer  rail  gains  a  diameter  enough  larger  than 
the  other  to  compensate  for  the  superior  length  of  the  outer 
rail,  although  a  theoretically  perfect  device,  is  gradually  going 
into  disuse.  To  be  effective  for  the  sharpest  curves,  the  coning 
must  be  so  great  as  to  produce  an  unsteady  motion  on  tan- 
gents, very  objectionable  at  high  speeds.  Moreover,  it  is  un- 
desirable to  seek  for  an  equilibrium  of  lateral  forces  in  a  car 
on  a  curve,  since  the  flanges  are  then  sure  to  strike  the  inner 
and  outer  rails  alternately  with  damaging  force,  as  that  equi 
librium  is  momentarily  disturbed.  It  is  far  better  that  the 
flange  should  press  steadily  against  the  outer  rail,  while  that 
pressure  is  modified  and  reduced  somewhat  by  the  elevation 
of  the  rail.  For  these  and  other  reasons,  car -wheels  are  now 
made  nearly  cylindrical. 


t 

LEVELLING.  181 


CHAPTER  VIII. 
LEVELLING. 

205.  The  field  operations  with  the  Engineer's  Level  are  of 
a  more  simple  character  than  those  performed  with  the  transit, 
yet  require  equal  skill  and  nicety  of  manipulation  in  order  to 
produce  trustworthy  results.     The  transit  is  used  to  ascertain 
the  relative  horizontal  position  of  points,  the  level  to  obtain 
their  relative  vertical  position. 

206.  In  order  to  express  the  elevation  of  points,  they  must 
be  referred  to  some  level  surface  of  known  (or  assumed)  eleva- 
tion; and  in  order  that  the  elevations  may  all  be  positive  up- 
wartl,  this  surface  of  reference  should  be  selected  below  all  the 
points  to  be  considered.    The  level  surface  of  reference  is  called 
the  datum. 

The  elevation  of  the  datum  is  always  zero.  The  elevation  of  any 
point  is  its  vertical  height  above  the  datum. 

Near  the  coast  the  sea  level  is  usually  adopted  as  the  datum; 
inland,  the  low  water  mark  of  a  river  or  lake,  etc. ;  but  it  is  not 
necessary  that  the  datum  should  coincide  with  a  water  surface. 
If  any  points  whose  elevations  are  to  be  ascertained  are  below 
the  water  surface,  the  latter  may  be  assumed  to  have  an  eleva- 
tion of  100  or  1000  feet  instead  of  zero;  that  is,  we  remove  the 
datum,  in  imagination,  to  100  or  1000  feet  below  the  level  of 
the  water  surface. 

20 7.  In  case  of  a  survey  commencing  at  a  point  quite  re- 
mote from  any  important  water  surface,  any  permanent  point 
may  be  selected  as  the  original  point  of  reference,  and  its  ele- 
vation maybe  assumed  at  100  or  any  other  number  of  feet; 
that  is,  we  fix  the  datum  at  the  same  number  of  feet  below  that 
point.     The  point  of  reference  is  called  a  bench,  or  bench- 
mark, and  is  designated  by  the  initials  B.M.     Other  benches 
are  established  at  intervals  during  a  survey,  and  their  eleva- 
tions determined  instrumentally.     They  are  then  convenient 


182  FIELD    ENGINEERING. 

points  of  known  elevation  for  future  reference.  We  cannot 
assume  the  elevation  of  more  than  one  bench  on  the  same  sur- 
vey, else  we  should  have  more  than  one  datum,  and  all  the 
results  would  be  thrown  into  confusion. 

208.  Htiving  established  the  first  bench  and  recorded  its 
elevation,  the  next  step  is  to  set  up  the  instrument  firmly  at  a 
moderate  distance  from  the  bench,  so  that  the  telescope  shall 
be  somewhat  higher  than  the  bench,  and  in  full  view  of  a  rod 
held  vertically  upon  it.     The  instrument  having  been  tested  for 
its  several  adjustments,  and  found  to  be  correct,  the  line  of  sight 
through  the  intersection  of  the  cross-hairs  is  known  to  be  hori- 
zontal when  the  bubble  stands  at  the  middle  of  its  tube.    Turn- 
ing the  line  of  sight  upon  the  rod,  the  point  of  the  rod  covered 
by  the  horizontal  cross-hair  is  known  to  be  on  a  level  with  the 
cross-hair;  and  the  latter  is  therefore  higher  than  the  bench  by 
the  distance  intercepted  on  the  rod  from  its  lower  end.    Add- 
ing this  distance  to  the  elevation  of  the  bench,  we  obtain  the 
elevation  of  the  cross-hair,  known  technically  as  the  "  Heiglit 
of  Instrument,"  and  designated  by  the  initials  .ZZi/. 

209.  The  distance  intercepted  on  a  rod  from  its  lower  end 
by  the  line  of  sight,  when  the  rod  is  held  vertically  on  any 
given  point,  is  called  the  reading"  of  the  rod  at  that  point. 

210.  Having  obtained  the  height  of  instrument,  the  eleva- 
tion of  any  point  somewhat  lower  than  the  cross-hair  is  easily 
ascertained  by  taking  a  reading  of  the  rod  upon  it.     The  read- 
ing subtracted  from  the  height  of  instrument  gives  the  eleva- 
tion of  the  point  above  the  datum.    The  elevation  of  any  num- 
ber of  other  points  may  be  similarly  obtained.    But  the  eleva- 
tion of  points  on  the  ground  higher  than  the  cross-hair,  or 
farther  below  it  than  the  length  of  the  rod,  cannot  be  deter- 
mined, because  in  either  case  the  line  of  sight  will  not  cut  the 
rod,  and  hence  there  can  be  no  reading.     In  order  to  observe 
such  points,  the  instrument  must  be  removed  to  a  new  posi- 
tion, higher  or  lower  than  before,  as  the  case  may  require. 


.  Before  the  instrument  is  removed  to  a  new  position, 
a  temporary  bench,  called  a  Turning  Point  (and  designated 
by  T.P.  or  "Peg")  must  be  established,  and  its  elevation  ascer- 


LEVELLING.  183 

tained  as  for  any  other  point,  but  with  more  care.  A  turning 
point  must  be  a  jinn  and  definite  point  whose  position  cannot 
readily  be  altered  in  the  least,  nor  lost  sight  of.  A  small  stake 
firmly  driven,  or  a  point  of  rock  projecting  upward,  is  fre- 
quently used.  The  reading  having  been  taken  on  the  turning 
point,  the  instrument  is  carried  forward  to  a  new  position, 
levelled  up  properly,  and  the  new  Height  of  Instrument  ob- 
tained by  a  new  reading  on  the  same  turning  point.  Since  the 
cross-hair  is  higher  than  the  point  (otherwise  there  could  be  no 
reading)  the  reading,  added  to  the  elevation  of  the  point,  gives 
the  Height  of  Instrument. 

212.  In  general,  the  intersection  of  the  cross-hairs  being 
higher  than  any  point  on  which  a  reading  is  taken: 

To  find  the  Height  of  Instrument,  add  the  reading  on  a  point 
to  the  elevation  of  the  point;  and 

To  find  the  Elevation  of  a,  point,  subtract  the  reading  on  it 
from  the  Height  of  Instrument. 

A  reading  taken  for  the  purpose  of  finding  the  Height  of 
Instrument  is  called  a  Backsight  (B.S\  A  reading  taken 
for  the  purpose  of  finding  the  elevation  of  a  turning-point  (or 
of  a  bench  used  as  such)  is  called  a  Foresight  (F.S).  Hence 
Backsights  are  always  plus,  and  Foresights  always  minus. 

213.  The  form  of  field-book  used  for  the  survey  of 
a  railroad,  or  other  continuous  line,  is  shown  below.    The  first 
column  contains  the  numbers  of  the  stations  on  the  line  and 
of  plus  distances  to  other  points  on  the  line  where  readings  are 
taken — also   the  initials  of    benches  and  turning  points,   in 
order,  as  they  occur.     The  second  column  contains  the  back- 
sights, taken  on  points  of  known  elevation  only.     The  third 
column  contains  the  height  of  instrument,  recorded  on  the 
same  line  as  the  elevation  of  the  turning  point  (or  bench)  from 
which  it  is  calculated.     The  fourth  column  contains  the  fore- 
sights, taken  on  new  turning  points,  and  benches  used  as  such, 
only.     The  fifth  column  contains  the  readings  taken  on  all 
other  points  noted  in  the  first  column.    The  sixth  column  con- 
tains the  elevations  of  all  points  observed.     The  right-hand 
page  is  reserved  for  remarks,  descriptive  of  the  benches  and 
their  location — of  objects  crossed  by  the  line,  as  roads,  streams, 
swamps,  ditches,  etc. ;  the  depths  of  streams,  etc. 


184 


FIELD    ENGINEERING. 


LEVEL  BOOK. 


Sta. 

B.S. 

H.I. 

F.S. 

Rod. 

EleV 

Remarks. 

B.M. 

4.683 

204.683 

2CO.OOO 

White  oak,  115  R. 

0 

2.1 

202.0 

1 

3.4 

201.3 

+  50 

5.2 

109.5 

Peg 

1.791 

197.260 

9.214 

195.469 

3.7 

193.6 

+"25 

7.0 

190.3 

Brook  5  wide  ;  1  deep 

+  50 

3.1 

194.2 

3 

0.5 

196.8 

Peg 

11.750 

208.574 

0.430 

196.824 

Pel 

11.933 

219.528 

0.979  | 

207.595 

+  90 

3.5 

21G.O 

4 

2.6 

21G.9 

B.M. 

2.075 

217.453 

Mapie,  78  L. 

5 

1.7 

217.8 

0 

0.9 

218.6 

Peg 

9.005 

227.801 

0.732 

218.796 

6.2 

221.6 

39.1G2 

11.361 

"*" 

When  a  bench  is  not  used  as  a  turning  point,  the  reading  on 
it  is  recorded  in  the  fifth  column. 

The  numbers  in  the  second,  fourth,  and  fifth  columns  come 
directly  from  the  rod,  those  in  the  third  are  obtained  by 
addition,  those  in  the  sixth  by  subtraction,  according  to  the 
rule  given  above.  The  additions  and  subtractions  made  on 
each  page  should  be  proved  before  proceeding  to  the  calcula- 
tions <»f  the  next.  When  correct,  the  difference  of  the  sums 
of  the  backsights  and  foresights  on  the  page  equals  the  differ- 
ence of  the  first  and  last  elevations  on  the  page.  Thus,  in  the 
form  given 

(39.162  -  11.361)  =  (227.801  -  200.000)  =  27.801 

In  this  proof  we  ignore  all  elevations  except  those  of  turn- 
ing points,  and  benches  used  as  such,  and  the  height  of  instru- 
hient. 

At  the  end  of  the  survey,  as  well  as  at  the  end  of  each  day's 
work,  a  bench  is  established  from  which  the  survey  may  be 
resumed  at  any  future  time  See  §§  28,  29,  and  80. 

214.  The  object  of  making  such  a  survey  with  level  and 
rod  is  to  furnish  a  profile  or  vertical  section  of  the  entire 
ine,  showing  in  detail  the  rise  and  fall  of  the  surface  over 


LEVELLING.  185 

which  it  passes.  The  profile  is  plotted  on  profile-paper  pub- 
lished for  the  purpose,  the  horizontal  scale  being  usually  400 
feet  to  an  inch,  and  the  vertical  scale  30  feet  to  an  inch.  This 
distortion  of  scale  magnifies  the  vertical  measures  so  that 
slight  changes  in  the  elevation  of  the  surface  may  be  seen 
distinctly. 

215.  When  only  the  difference  of  level  of  two  extreme 
points  is  required,  the  survey  is  more  simple.     No  readings 
are  taken  except  on  turning-points,  the  backsights  and  fore- 
sights being  recorded  in  separate  columns.     No  calculation  is 
required  until  the  survey  is  finished,  when — the  first  reading 
having  been  taken  on  one  of  the  given  points,  and  the  last  on 
the  other — the  difference  of  the  sums  of  the  backsights  and 
foresights  is  the  difference  in  elevation  of  the  two  points,  ac- 
cording to  the  method  of  proof  mentioned  in  §  213.     Thus  the 
difference  in  level  of  any  two  benches  established  on  a  previ- 
ous survey  may  be  tested,  and,  if  found  correct,  all  the  inter- 
mediate elevations  on  the  line  may  be  assumed  to  be  correct 
also.     The  discrepancy  should  not  exceed  one  tenth  of  a  foot 
in  any  case,  and  is  usually  much  less. 

216.  Any  lack  of  adjustment  in  the   instrument  gives 
the  line  of  sight  a  slight  angle  of  elevation  or  depression, 
causing  a  slight  error  in  every  reading,  proportional  to  the 
distance  of  the  rod  from  the  instrument.    But  the  errors  being 
equal  for  equal  distances,  and  the  backsights  and  foresights 
having  opposite  signs  in  our  calculations,  the  errors  cancel 
when  the  distances  are  equal.     Hence,  to  avoid  errors  in  ele- 
vation, each  new  turning-point  should  be  as  nearly  as  possible 
at  the  same  distance  from  the  instrument  as  the  point  on  which 
the  last  backsight  was  taken.      For  precise  reading,  the  rod 
should  not  be  more  than  400  feet  from  the  instrument. 

217.  Another  cause  of  error  in  readings  is  want  of  verti- 
cality  in  the  rod.     This  may  be  avoided  by  the  use  of  a  disk- 
level,  or  in  the  absence  of  wind,  by  balancing  the  rod.     The 
rod  may  be  plumbed  one  way  by  the  vertical  cross-hair  of  the 
level,  and  to  ensure  a  vertical  reading  in  the  plane  of  the  line  of 
sight,  the  rod  may  be  gently  waved  each  side  of  the  vertical 
toward  and  from  the  instrument,  the  shortest  reading  being 


186 


FIELD  ESTGIKEERIKG. 


the  correct  one;  or  in  case  of  a  target  rod,  the  target  should 
rise  to,  but  not  above  the  horizontal  cross-hair,  as  the  rod  is 
waved. 

218.  When  very  long  sights  are  required  to  be  taken  with 
the  level,  another  source  of  error  must  be  considered,  namely, 
the  curvature  of  the  earth. 

A  level  line  is  parallel  to  a  great  circle  of  the  earth,  and  is 
therefore  an  arc  of  a  circle,  or  may  be  so  considered. 

A  horizontal  line  is  a  straight  line  parallel  to  the  plane  of  the 
horizon.  Therefore  the  line  of  sight,  being  a  horizontal  line, 
is  tangent  to  the  circle  of  a  level  line  passing  through  the  in- 
strument. 

To  find  the  correction  in  elevation  du&  to  curvature  of  tlie 
earth  for  any  distant  station.  Fig.  82. 


FIG.  82. 

Let  A  be  the  station  of  the  instrument  /,  and  B  the  distant 
station  observed. 

Let  R0  —  CI—  the  radius  of  curvature  of  the  earth,  or  of  the 
parallel  arc  ID.  Let  L0  =  ID  =  the  level  distance  between 
A  and  B.  Then  IE,  perpendicular  to  CI,  is  the  line  of  sight, 
BE  is  the  reading  of  the  rod,  and  DE  =  E0  —  the  correction 
due  to  curvature. 

By  Tab.  I.,  24,  IE*  =  DE  (DE+2JRJ;  but  since  DE  is 
very  small  compared  with  2.Z?C,  it  may  be  omitted  from  the 
parenthesis,  and  since  IE  =  ID  =  L0  very  nearly,  because 
the  angle  ACB  is  very  small,  we  have  L*  =  2R0E0. 

E=^-  (315) 


E0  is  to  be  added  to  the  apparent  elevation  of  station  B. 


LEVELLING.  187 

219.  Refraction.      In  observing  distant  stations   the 
line  of  sight  passing  through  the  atmosphere  is  refracted  from 
the  straight  line  IE,  Fig.  82,  and  takes  the  form  of  a  curve, 
which,  for  practical  purposes,  may  be  considered  as  the  arc  of 
a  circle,  concave  downwards.     Its  radius,  depending  on  the 
conditions  of  the  atmosphere,  varies  from  5i  to  1\  times  the 
radius  of  curvature  of  the  earth.     1R0  is  considered  a  good 
average  value. 

Refraction  causes  the  observed  object  to  appear  too  high, 
while  the  curvature  of  the  earth  causes  it  to  appear  too  low; — 
the  effects  being  contrary,  the  correction  for  curvature  is  re- 
duced by  the  correction  for  refraction.  If  we  let  H0  —  the 
total  correction  for  both  curvature  and  refraction,  to  be  added 
to  the  apparent  elevation  of  the  observed  object,  then 

H»=\E°=^f  (316> 

Table  XVII.  is  calculated  by  this  formula,  assuming  a  mean 
value  of  R0  =  20,913,650  feet. 

220.  The  form  of  the  earth  is  approximately  an  el- 
lipsoid of  revolution.    Its  meridian  section  at  the  mean  level 
of  the  sea  is  an  ellipse,  the  semi-axes  of  which  are,  according 
to  Clarke, 

at  the  equator  A  =  6378206  metres  [6.8046985] 
at  the  poles       £=6356584      "       [6.8032238] 

According  to  the  same  authority 

1  metre  =  3.280869  feet  [0.5159889] 

Therefore  the  semi-axes  expressed  in  feet  are 

A  =  20  926  058  feet  [7.3206874] 

B  =  20  855  119    "  [7.3192J.27] 

Then  the  radius  of  curvature  of  the  meridian 

at  the  equator,  ~  =  Eo  =  20  784  422  ft.  [7.3177379] 
at  the  poles,      ~  =  J?0  =  20  997  240  "  [7.3221622] 


188  FIELD   ENGINEERING. 

In  latitude  40°  the  radius  of  curvature  of  the  meridian  is 
20  871  900,  and  of  a  section  at  right  angles  to  the  meridian, 
20  955  400;  the  mean  value,  or  R0  =  20  913  650  [7.320430],  be- 
ing adopted  for  general  use.  The  error  in  the  correction  HQ 
eq.  (316)  due  to  this  assumption  will  usually  be  much  less  than 
that  due  to  the  assumed  value  of  the  radius  of  refraction. 

221.  Levelling  by  Transit  or  Theodolite.  When 
a  transit  has  a  level-tube  attached  to  the  telescope,  it  may 
be  used  as  a  Theodolite  for  levelling,  and  for  taking  vertical 
angles.  If  the  instrument  be  in  perfect  adjustment,  the  line 
of  sight  will  be  horizontal  when  the  bubble  stands  at  the 
middle  point  of  the  tube,  and  the  reading  of  the  vertical  circle 
will  be  zero.  Should  there  be  a  small  reading  when  the  line  of 
sight  is  horizontal  it  is  called  the  index  error.  When  the  line 
of  sight  is  not  horizontal,  the  angle  which  it  makes  with  the 
plane  of  the  horizon  is  called  an  angle  of  elevation,  or  of  de- 
pression, according  as  the  object  upon  which  the  line  of  sight 
is  directed  is  above  or  below  the  telescope.  This  angle  is 
measured  on  the  vertical  circle,  being  the  difference  of  the 
reading  and  the  index  error,  when  both  are  on  the  same  side 
of  the  zero  mark,  and  their  sum,  when  they  are  on  opposite 
sides.  When  the  distance  to  an  observed  object  is  known, 
and  its  angle  of  elevation  or  depression  is  measured,  we  may 
calculate  its  vertical  height  above  or  below  the  telescope. 

.  (  elevation 
Let  ±  a  —  angle  of  -   _ 

f  depression 

"        L  =  the  horizontal  distance 
"       L'  =  the  distance   parallel  to 

line  of  sight 
"        h  =  difference  in  elevation  of 

object  and  instrument. 

Then  for  short  distances, 
\  h  =  L  tan  a  =  L'  sin  a      (317) 

FIG.  83.  For  long  distances  the  curvature  of 

the  earth  and  refraction  must  be  considered.     Fig.  83. 

Let  /  be  the  place  of  the  instrument,  and  F  the  object 
observed. 


LEVELLING.  189 

Let  L0  =  the  distance,  measured  on  the  chord  of  the  level 
arc  ID,  passing  through  the  instrument;  and  let  ty  =  the 
number  of  seconds  in  the  arc  ID  •  hence,  since  for  ordinary 
distances  the  chord  and  arc  are  sensibly  equal, 

^  =  LR    306264"-8  [5.314425] 

or  giving  to  R0  its  mean  value,  §  220, 

if^  =  L0X  .0098627  [7.993995] 

or  a  fraction  less  than  1"  per  100  feet. 

Let  IF  be  the  arc  of  the  refracted  ray,  and  assuming  that  its 
radius  is  7R0,  the  arc  will  contain  }th  the  number  of  seconds 
of  the  arc  IF 

IF',  tangent  to  IF,  is  the  direction  of  the  telescope;  IF  is 
the  chord  of  the  arc  IF,  and  IE  is  the  horizontal. 

Let  a  =  EIF'  =  observed  angle  of  elevation.  Then  EIF  '  — 
true  angle  of  elevation  =  EIF'  —  F'IF=  a  —  £  .  \ip  —  a  — 
.071^. 

The  angle  EID  =  $i/>  .-'.  DIP  =  &f>  +  a  -  .071^;  and 
IDF  =  90°  +  W  .  .'.  IFD  =  90°  -  (#  -f  a  -  .071^). 

"We  now  solve  the  triangle  IFD  for  the  side  DF  =  h,  and 
find 

sin  (ifl  +  g-.  0710 

^°  cos        -- 


For  an  observed  angle  of  depression  make  a  negative  in  the 
formula. 

The  coefficient  .071  is  called  the  coefficient  of  refraction,  this 
being  a  fair  average  value,  while  its  extreme  range  is  from  .067 
to  .100  under  varying  conditions  of  the  atmosphere,  and  values 
of  the  angle  a. 

When  the  difference  in  elevation  of  two  or  more  distant 
objects  is  required,  we  obtain  the  elevation  of  each  separately, 
and  subtract  one  elevation  from  another.  The  elevation  of  the 
observed  object  is  given  by  (//.  /.)  ±  h. 

222.  To  find  the  Height  of  Instrument  of  a  transit  or 
t/ieodoUte  by  an  observation  of  the  horizon.  Fig.  84. 


190 


FIELD   ENGINEERING. 


Let  I  be  the  place  of  the  instrument,  and  let  a  =  observed 
angle  of  depression  of  the  horizon. 

Let  F  be  the  point  where  the  refracted  ray  meets  the  level 
surface,  and  draw  the  chords  IF  and  AF. 

Let  iff  =  the  angle  ACF,  let  h  =  AI,  and  let  k  =  the  coeffi- 
cient of  refraction. 

In  the  triangle  IAF, 

IAF  =  90°  +  i#,  AFI  =  iif>  -  faf},   AIF  =  90°  -  (iff  -  kip] 
Hence  FIE  =  if>  —  k$.    But  FIE  =  a  -f  faj> 

^=__^_  (319) 

Let  F"  be  the  tangent  point  of  a  right  line  drawn  through  /; 

E 


FIG.  84 


then  AI  =  CF'  exsec  ACF",  but  CF"  =  R0,  and,  since  $  is 

1  —  & 
always  very  small,  ACF'  =  #$  +  a)  very  nearly  =   — .  « 


=  ft  exsec  = XT  a 


(320) 


I -2k 

Giving  to  E0  its  mean  value,  §  220,  and  assuming  k  =  TV 
log  7*  =  7.320430  -f  log  exsec  1.0801  a  (321) 


LEVELLING.  191 

Otherwise,  we  may  solve  the  triangle  AIF  since 
AF  =  2B0  sin  &  =  2R0  sin  ^  "  ^ 

sin(i0-*0) 
and  fc_.__ 


cos  Y 
When  k  =  TV 


h  =  2Rn  sin  A  a  .  -  (323) 

cos  if  a 

Example. — The  observed  dip  of  the  sea  horizon  is  24'  =  a- 
What  is  the  height  of  the  instrument  above  thejsea? 

By  eq.  (321)  1.0801  X  a  X  60  =  1555". 34  3.191825 

2 


6.383650 

Table  XXVI.     (q  -  2  0        9. 070130 
R0  7.320430 

h  =  594.58  2.774210 

Methods  of  determining  heights  by  distant  observations  can- 
not be  relied  on  for  more  than  approximate  results,  since  they 
necessarily  involve  the  uncertain  element  of  refraction,  and 
usually  a  lack  of  precision  in  the  vertical  angle,  the  arc  reading 
only  to  minutes  in  ordinary  instruments.  These  methods,  how- 
ever, are  useful  where  no  great  accuracy  is  required,  as  for  a 
temporary  purpose  until  levels  can  be  taken  in  the  regular  way, 
or  for  interpolating  between  points  of  established  elevation. 

223.  Stadia  Measurements. 

It  is  sometLnes  convenient  to  determine  distances  by  instru- 
mental observation  For  this  purpose  two  additional  cross- 
hairs may  be  placed  in  the  telescope  parallel  to  each  other  and 
equidistant  from  the  central  cross-hair.  These  are  called  stadia 
hairs,  and  distances  determined  by  them  are  called  stadia 
measurements.  The  stadia  hairs  are  adjusted  so  as  to  inter- 
cept a  certain  space  on  a  rod  held  at  a  certain  distance  from 
the  instrument  and  perpendicular  to  the  line  of  sight.  For  any 


192 


FIELD   ENGINEERING. 


other  place  of  the  rod,  the  distances  and  intercepted  spaces 
are  nearly  proportional.  The  exact  relation  is  given  below. 
Fig.  85. 

Let  I  —  AB,  the  distance  of  the  rod  from  the  vertical  axis 
of  the  instrument;  c  =  the  distance  from  the  axis  to  the  ob- 
ject glass  of  the  telescope;  a  =  the  distance  from  the  object- 


FIG. 


glass  to  the  rod ;  i  =  the  space  between  the  stadia  hairs ;  s  = 
CD  the  space  intercepted  by  them  on  the  rod;  and/  =  the 
focal  distance  of  the  object-glass.  We  then  have  by  optics, 

'-  =  — ~-f  whence  a  —f=-.s;  and  since  a  =  I  —  c  .  \  I  — 

(/-|-  c)  =  -s.    Now  in  any  given  instrument  the  focal  distance 

/,  and  the  space  between  the  stadia  hairs  i  are  constant,  while 
s  and  c  vary  with  I.  For  any  other  distance  I',  we  then  have 


I'  —  (/+  c')  =  -s',  and  combining  the  two  equations 


(324) 


s  '  is  usually  assumed  at  1  foot  and  I'  —  (/  +  c')  at  100  feet, 
and  the  stadia  hairs  are  then  adjusted  accordingly.  The  focal 
distance /may  be  found  by  removing  the  object  glass  and  ex- 
posing it  to  the  rays  of  the  sun  and  noting  at  what  distance 
from  the  surface  of  the  lens  the  rays  form  a  perfect  and  min 
ute  image  of  the  sun  on  a  smooth  surface;  the  distance  c'  is 
measured  on  the  telescope  when  the  rod  is  clearly  in  focus, 
at  the  assumed  distance. 

To  measure  any  other  distance,  the  rod  is  again  observed 
at  the  desired  point,  and  the  space  s  noted,  which,  placed  in 
eq.  (324),  gives  I  —  (f-\-  c)  at  once.  We  then  measure  c  on 
the  telescope,  and  adding  (/  -f-  c),  obtain  I,  the  distance  re- 
quired. 


LEVELLING.  193 

But  inasmuch  as  c  has  but  a  small  range  of  values,  it  will 
usually  be  sufficient  to  assume  for  it  a  mean  value,  as  a  con- 
stant. In  this  case  we  may  find  the  value  of  (/  -f-  c)  =  IF 
for  the  instrument  used.  Making  c  =  c  in  eq.  (324),  and  solv- 
ing for  (/-f-  c),  we  have 

=?  (325) 


and  by  laying  off  on  level  ground  any  two  distances  from  the 
instrument  for  I'  and  I,  as  100  and  500,  and  observing  the 
corresponding  spaces  s'  and  s  intercepted  on  a  rod,  we  insert 
them  in  eq.  (325)  and  find  (/+  c). 

Having  found  (/+  c),  lay  off  (100  +/+  c)  from  the  instru- 
ment and  adjust  the  stadia  hairs  to  inclose  just  one  foot  on 
the  rod  at  that  distance.  Any  other  distance  is  then  found  by 
the  formal  a, 

(326) 


Example.—  At  I'  =  100  we  finds'  =  1.00,  and  at  I  =  500  we 
find  s  =  5.061. 

Hence,  eq.  (325)       /+  c  =  ^  =  1-502 


and  eq.  (326)  I  =  100  s  +  1.5;  provided  the  stadia  hairs  be  ad- 
justed so  as  to  intercept  1  foot  at  101.5  feet  distance  from  the 
centre  of  the  instrument. 

224.  The  foregoing  formulae  are  all  that  are  necessary  for 
horizontal  sights,  but  since  the  line  of  collimation  is  generally 
inclined  more  or  less  to  the  horizon,  it  follows  that  the  stadia 
hairs  will  intercept  a  larger  space  on  the  vertical  rod  than 
that  due  to  the  true  horizontal  distance.  We  therefore  require 
a  formula  for  reducing*  inclined  measurements 
to  the  horizontal.  Fig.  86. 

Let  a  =  EFG  =  the  angle  of  inclination  of  the  -line  of  colli- 

mation IG  ; 

"    0  =  CFD  --  the  visual  angle  defined  by  the  stadia  hairs; 
"    s  =  CD  =  space  intercepted  on  a  vertical  rod. 
Then  (Fig.  85), 


tan  *=          =  !.  (327) 


194  FIELD    ENGINEERING. 

Ill  Fig.  86 

«  =  CE  —  DE  =  EF  [tan  (a  -f  $6)  -  tan  (a  -  £ 
while  the  true  value  (for  the  same  distance)  would  be 


Dividing  one  by  the  other  we  derive 
C'D'  2  tan 


s  tan  (a  -f-  £0)  —  tan  (a  —  $0) 

By  giving  to  «'  and  I'  —  (/-j-  c)  in  eq.  (327)  their  customary 


-ID' 


FIG.  86. 

values,  viz.,  1  and  100,  we  have  tan $0  =  .005    .-.  0  =  34'  22". 63 
and  by  Trig.  Table  II.  70, 

tan  (a  4--J-0)  —  tan  (a  —  ¥J)  =  — - — — - 

cos  (a  -j-  |0)  cos  (a— £&) 

Since  0  is  small,  we  have  sensibly 

sin  0  =  2  tan  £0,  and  cos  (a  -|-  £0)  cos  (a  —  £6)  =  cos2  a 
and  the  last  equation  reduces  sensibly  to 

SlS-^&^a  (328) 

which  is  the  coefficient  of  reduction  required  by  which 
to  multiply  the  observed  space  s  in  case  of  inclined  sights. 

Hence  the  formula  for  distance  (eq.  326)  becomes  in  this  case 
without  sensible  error 

I  =  100  s  cos2  a  +  (/+  c)  (329) 

Tables  XVIII.  and  XIX.  Lave  been  calculated  by  the  exact 
formula  for  the  coefficient. 


LEYELLlKG.  195 

Example.— Given  :    a  =  8°  20'  and  s  =  9.221;  what  is  the 
horizontal  distance  to  the  rod? 

Eq.  (329)    100  log.  2. 

s      9.221  "  0.964778 

a  8°  20'  Tab.  XIX.     "  9.990780 

902.7  2.955558 

f-\-c      1.5       .'.  Aw.  904.2ft. 


The  rod  man  should  have  a  disk  level  to  insure  keeping  the 
rod  vertical. 

225.  Another  method  of  procedure  is  that  in  which 
the  rod  is  always  held  perpendicular  to  the  line  of  collimation, 
however  much  inclined  the  latter  may  be.  To  secure  this  posi- 
tion of  the  rod,  a  small  brass  bar  is  attached,  having  sights 
upon  it  through  which  the  rodman  watches  the  instrument 
during  an  observation,  the  line  of  sight  being  at  right  angles  to 
the  rod.  The  distance  thus  obtained  is  of  course  parallel  to 
the  line  of  collimation,  and  requires  to  be  reduced  to  the  hori- 
zontal. 

For  this  purpose,  we  have  (Fig.  87). 


FIG.  87. 

IE  =  IG  cos  a  -f-  BG  sin  a 
or  IE  =  (100  s + /+  c)  cos  a  -f-  r  sin  a  (330) 

in  which  r  is  the  reading  of  the  rod  by  the  line  of  collimatioa 
For  the  elevation  of  the  point  B  above  /, 

EB  =  HG  -  GB  cos  a 
or  EB  =  (100  s  +/+  c)  sin  a  -  r  cos  a  (331) 


FIELD    EJ^GISTEERI^G. 

When  the  distances  are  sufficiently  great,  correction  must  be 
made  for  curvature  of  the  earth  and  refraction,  as  already  ex- 
plained. 

This  method  is  employed  by  the  topographical  parties  of  the 
U.  S.  Coast  Survey  in  connection  with  the  plane  table.  Their 
instruments,  however,  arc  so  constructed  as  to  give  distances 
in  metres,  and  heights  in  feet,  requiring  a  modification  of  the 
above  formulae. 


CHAPTER    IX. 

CONSTRUCTION. 

226.  The  engineer  department  of  a  railway  com- 
pany is  usually  reorganized  for  the  construction  of  the  road, 
as  follows  :  Chief  engineer,  Division  engineers,  Resident 
engineers, -Assistant  engineers.  On  some  roads  the  division 
engineers  are  styled  "Principal  Assistants;"  the  resident 
engineers,  "Assistants;"  and  the  assistant  engineers  are  de- 
signated according  to  their  duties,  as  "leveller,"  "  roclman," 
etc. 

A  resident  engineer  has  charge  of  a  few  miles  of  line, 
limited  to  so  much  as  he  can  personally  superintend  and 
direct.  He  has  one  or  more  assistants  and  an  axman  in  his 
party.  All  instrumental  work  is  done  and  all  measurements 
taken  by  the  resident  engineer  and  his  assistants. 

A  division  engineer  has  charge  of  several  residencies, 
and  inspects  the  progress  of  the  work  on  his  division  once 
or  twice  a  week.  In  his  office,  which  should  be  centrally 
located,  all  maps,  profiles,  plans,  and  most  of  the  working 
drawings  required  on  his  division  are  prepared.  To  him  the 
resilient  engineers  make  detailed  reports  once  a  month,  or 
oftener  if  necessary,  which  he  passes  upon  as  to  their  cor- 
rectness, and  from  which  he  makes  up  a  monthly  report,  or 
estimate,  of  the  amount  and  value  of  the  work  done  and  ma- 
terials provided  by  each  contractor  on  his  division.  The  esti- 
mates are  forwarded  about  the  first  of  each  month  to  the 
chief  engineer,  who  examines  and  approves  them,  returning 
for  modification  any  that  seem  to  require  it. 


CONSTRUCTION.  197 

The  chief  engineer  lias  charge  of  the  entire  work, 
and  directs  the  general  business  of  the  engineer  department, 
lie  occasionally  inspects  the  work  along  the  line. 

227.  Clearing  and   Grubbing'.     The  first  step   in 
the  work  of  construction  is  to  clear  off  all  growth  of  timber 
within  the  limits  of  the  right  of  way.     The  resident  engineer 
with  his  party  passes  over  the  line,  making  offsets  to  the  right 
and  left,  and  blazing  the  trees  which  stand  on,  or  just  within, 
the  limits  of  the  company's  property.      The  blazed  spot  is 
marked  with  a  letter  0,  as  a  guide  to  the  contractor.     After 
felling,  the  valuable  timber  should  be  piled  near  the  boun- 
dary lines,  to  be  saved  as  the  property  of  the  company.     The 
brushwood  is  burned. 

Where  a  deep  cut  is  to  be  made,  the  stumps  are  left  to  be 
removed  as  the  earth  is  excavated.  In  very  shallow  cuts  and 
fills  the  contractor  will  generally  prefer  to  tear  up  the  trees 
by  their  roots  at  once,  rather  than  to  grub  out  the  stumps 
after  clearing.  Where  the  embankments  will  be  over  three 
feet  high,  grubbing  is  not  necessary;  but  the  trees  require  to 
be  low-chopped,  leaving  no  stump  above  the  roots.  The  engi- 
neer should  indicate  to  the  contractor  the  localities  where  each 
process  is  suitable. 

228.  While  the  clearing  is  in  progress,  the  engineer  should 
run  a  line  of  test  levels  touching  on  all  the  benches  to  verify 
their  elevations  ;  he  may  also  rerun  the  centre  line,  replacing 
any  stakes  that  may  have  disappeared,  and  setting  guard  plugs 
to  any  important  transit  points  which   may  not   have  been 
previously  guarded.     If  any  changes  in  the  alignment  have 
been  ordered,  these  may  be  made  at  the  same  time. 

220.  Cross  Sections.  The  resident  engineer  is  fur- 
nished with  a  profile  of  the  portion  of  the  line  in  his  charge, 
upon  which  is  plainly  indicated  by  line  and  figures  the  estab- 
lished grade.  From  this  he  calculates  the  elevation  of  grade 
at  each  station,  and  by  subtracting  this  from  the  elevation  of 
the  surface,  he  derives  the  depth  of  cut  or  fill  (-f-  or  — )  to  be 
made  at  each  point.  The  grade  given  on  the  profile  is  that 
which  is  subsequently  called  the  subgrade,  being  the  surface 
of  the  road-bed.  The  final  or  true  grade  is  the  upper  surface 
of  the  ties  after  the  track  is  laid. 


198  FIELD   ENGINEERING. 

The  base  of  a  cross  section  is  identical  with  the  width  of  the 
road-bed.  It  is  made  wider  in  cuts  than  in  fills  to  allow  for 
the  side  ditches.  Six  feet  should  be  allowed  in  earth,  and 
four  feet  in  rock  cuts.  The  ratio  of  the  side  slopes 
depends  upon  the  material.  The  usual  slope  ratio  for  earth  is 
1|  horizontal  to  1  vertical  for  both  excavation  and  embank- 
ment. Damp  clay  and  solid  gravel  beds  will  stand  for  a  time 
in  cuts  at  1  to  1,  or  an  angle  of  45°,  but  this  cannot  be  perma- 
nently depended  on.  On  the  other  hand,  fine  sand  and  very 
wet  clay  may  require  slopes  of  If  to  1  or  2  to  1.  Exceptional 
cases  require  slopes  of  3  or  4  to  1.  In  rock  work  the  slopes  are 
usually  made  at  £  to  1  for  solid,  |  to  1  for  loose,  and  1  to  1  for 
very  loose  rock,  liable  to  disintegrate.  Rock  embankments 
stand  at  1  to  1. 

23O.  All  cross  sections  are  taken  in  Vertical  planes  at 
right  angles  to  the  direction  of  the  centre  line.  Figs.  88,  89. 
Formulae. 

Let  b  =  AB,  the  base  of  section,  or  road-bed. 
"    s  =  =  the  slope  ratio 

"  d  =  CO  =  the  cut  (or  fill)  at  the  centre  stake. 
"  h  =  DHor  EN=  the  cut  (or  fill)  at  the  side  stake. 
"  x  =  CD  =  the  "distance  out"  from  centre  to  side  stake. 
"  y  =  h-d  =  KD. 
We  have  at  once  from  the  figures  the  general  formula 

x=$b-\-8h  (332) 

When  the  ground  is  level  transversely; 

h  —  d,  and  x  =  $b  -f-  sd. 

For  embankment  use  the  same  formula,  considering  d  or  h  as 
positive  in  this  case  also,  the  figure  being  simply  inverted. 
When  the  ground  is  inclined  transversely; 

h  =  CO  -j-  DK  —  d  -f-  y    on  the  upper  side  in  cuts; 

x=\b  +  sd-\-sy  (333) 

and  h  =  EN =d  —  y    on  the  lower  side  in  cuts 

•*.  *  =  \b  -f  sd  -  sy  (334) 


CONSTRUCTION.  199 

"For  embankments  use  the  same  formulae,  but  apply  eq.  (333)  to 
the  lower  side  and  eq.  (334)  to  the  upper  side,  the  figure  being 
inverted.  The  points  D  and  E  on  the  ground  are  usually  found 
by  trial,  such  that  the  corresponding  values  of  x  and  y  will 
verify  the  formulae. 

When  the  natural  slope  FD  or  LE  is  uniform  its  ratio  s'  may 
be  found  by  measuring  along  the  section  the  horizontal  dis- 
tance necessary  to  change  the  reading  of  the  rod  1  foot  (or  halt 
the  distance  necessary  to  change  it  2  feet,  etc.).  Then,  having 
found  the  depths  of  cut  (or  fill)  at  J^and  L,  distant  %b  from  the 
centre  C,  we  have 

BE  =  sh  =  s'(h  -  BF) 
and  AN  =  s?i  =  s'(AL  -  h) 

From  these  we  have,  for  the  upper  side  in  cuts,  and  lower  side 
in  fills. 

h  =  y-*^  BF  .  •.  x  =  \b  +  -£--  BF          (335) 
also,  for  the  lower  side  in  cuts,  and  upper  side  in  fills, 

h  =  yVj  AL  •'•  x  =  ^b  +  7^77  AL 
We  also  have 


and  (337) 


whence  the  points  D  and  E  may  be  found  by  the  level. 

But  points  D  and  E  thus  calculated  should  have  their  posi- 
tions verified  by  the  general  formula,  eq.  (332),  lest  the  slope 
s'  may  not  have  been  perfectly  uniform. 

When  the  natural  surface  intersects  the  base  between  the 
points  A  and  B,  the  section  is  said  to  be  in  side  hill  work, 
Fig.  90.  Both  portions  of  the  section  are  then  determined  by 
eq.  (333),  or  where  the  slope  s'  is  regular,  by  eq.  (335)  measuring 
in  every  case  from  the  centre  stake  C;  but  observing  that 
when  the  centre  is  in  cut  and  one  side  in  fill,  or  vice  versa,  that 
d  must  be  considered  negative  for  that  side,  whence  eq.  (333) 
becomes  for  this  case 

x  =  ib  -  sd  -f  *y  (333)' 


200 


FIELD   EKGINEEKLtfG. 


231.  Staking-  out  Earthwork.  Beginning  at  a 
point  on  the  centre  line  where  the  grade  cuts  the  natural  sur- 
face, the  engineer  drives  a  grade  stake  (marked  0.0)  and  notes 
the  point  in  the  cross-section  book.  If  the  line  of  intersection 
of  the  road-bed  and  surface  would  make  an  acute  angle  with 
the  centre  line,  he  also  finds  the  points  where  the  edges  of  the 
proposed  road-bed  will  intersect  the  surface,  drives  grade 
stakes,  and  also  stakes  out  a  cross  section  through  each  of 
those  points,  if  necessary. 

Then  advancing  to  the  next  point  on  the  centre  line  where 
a  section  is  required,  he  finds  its  elevation  with  the  level  (veri- 
fying or  correcting  the  elevation  taken  on  the  location),  calcu- 
lates the  depth  of  cut  or  fill  CG,  which  is  then  marked  upon 
the  back  of  a  stake  there  driven;  a  cut  being  designated  by  G 
and  a  fill  by  F. 

If  the  ground  is  level  transversely  (Fig.  88),  ha  calculates  x  by 


eq.  (382)  and  lays  off  this  distance  at  right  angles  to  the  centre 
line,  driving  slope  stakes  at  the  points  D  and  E,  marked  with 
the  depth  of  cut  or  fill.    The  marked  side  of  slope  stakes  should 
face  the  centre  line. 
If  the  ground  is  inclined  transversely  (Fig.  89),  he  first  measures 


FIG. 


the  distance,  |&,  to  F,  and  finds  the  depth  BFior  record.  He 
then  proceeds  to  find  the  point  D.  If  the  natural  slope  be  uni- 
form, D  may  be  found  by  eq.  (335)  or  (337),  verifying  the  result 
by  eq.  (332).  The  point  E  of  the  other  slope  may  be  found 
similarly,  using  eq.  (336)  or  eq.  (337);  verifying  by  eq.  (332). 


CONSTRUCTION.  201 

232.  If  the  ground  be  irregular,  the  depth  of  cut  or  fill  is 
found  not  only  at  the  centre  and  edges  of  the  road-bed,  but 
also  at  every  other  point  along  the  cross  section  where  the  sur- 
face slope  changes,  all  of  which  depths  are  recorded,  together 
witk  their  respective  distances  from  the  centre.     To  find  the 
point  D :  assume  a  point  supposed  to  be  near  D,  and  there 
take  a  reading  of  the  rod.     The  difference  of  the  readings  at 
that  point  and  at  C  equals  y'  for  that  point,  which  inserted  in 
eq.  (333)  gives  a  value  x.     If  x  agrees  with  the  horizontal  dis- 
tance of  the  assumed  point  from  C,  the  true  position  of  D  has 
been  found.     If  x'  be  greater  than  this,  by  subtracting  the  eq. 
x  =  ib  -\-  sd  -\-  sy'  from  eq.  (333)  we  derive 

X  =  X>+S(y-y")  (338) 

the  last  term  of  which  shows  the  correction  to  be  added  to  x '. 
Now  in  advancing  from  the  assumed  point  to  the  extremity  of 
x',  the  rise  of  the  surface  is  approximately  (y  —  y'},  and  if,  in 
going  the  additional  distance,  s(y  —  y'},  a  further  rise  is  en- 
countered, this  last,  multiplied  by  s,  must  also  be  added  to  x' , 
and  so  on  until  the  additional  advance  makes  no  change  in  the 
value  of  y.  The  point  thus  found,  verified  by  eq.  (332),  is  the 
point  D  required. 

But  if  x'  be  less  than  the  distance  of  the  assumed  point  from 
C,  we  have 

x  =  x'-s(y'  -y}  (338)' 

the  corrections  being  subtractive. 

The  point  E  on  the  other  slope  is  found  in  a  similar  manner, 
using  eq.  (334)  for  the  value  of  x  ;  if  x'  be  greater  than  the  as- 
sumed distance,  we  have 

x  =  x  -s(y-  y'}  (339) 

the  corrections  being  subtractive  ;  but  if  x'.  be  less  than  the  as- 
sumed distance, 

*=*'  +  «(y'-y)  (339)' 

the  corrections  being  additive. 

233.  In  side-hill  work  (Fig.  90)  proceed  in  the  same 
manner,  using  eqs.  (333)  or  (333)'  and  (338)  in  all  cases  of  un- 
even ground.     When  the  surface  slope  s'  is  uniform,  eq.  (335) 
may  be  used,  if  preferred,  on  either  side.     In  addition  to  the 


202 


FIELD 


centre  and  side  stakes,  a  grade  stake  is  driven  at  the  point  0, 
where  the  surface  intersects  the  grade,  the  stake  facing  down 
hill. 

To  find  a  grade  point,  set  the  target  to  a  reading  equal  to  the 
height  of  instrument  less  the  elevation  of  grade,  and  stand  the 
rod  at  various  points  along  the  given  line  until  the  target  coin- 
cides with  the  line  of  collimation. 


FIG.  90. 

234.  When  two  materials  are^foundin  the  same  section, 
as  rock  overlaid  with  earth,  each  material  "requires  its  own 
slope,  and  a  compound  section  is  the  result.  To  stake 
out  work  of  this  description,  the  depth  of  earth  to  the  rock  must 
be  known,  and  may  be  nearly  ascertained  by  reference  to  an 
adjacent  section  already  excavated.  Fig.  91. 


Then 


h— 


Let  ai  be  the  depth  of  earth  at  C 
"   a*      "  "  "       POT 

"    Si  be  the  ratio  of  rock  slope 
"    s3      "  "       earth  slope 

s,(d  —  a i  ±  y,)  -j-  «2(a2  ± 


(340) 


in  which  y\  —  difference  of  rod  readings  on  the  rock  at  C0  and 
Di,  or  C0  and  EI ;  and  y.z  —  difference  of  rod  readings  on  the 
surface  at  P  and  D2,  or  at  Q  and  E*.  The  upper  sign  applies 
to  the  upper  side,  the  lower  sign  to  the  lower, 


CONSTRUCTION.  203 

It  is  better,  however,  to  make  an  indefinite  cross  profile  at 
first,  driving  two  reference  stakes  quite  beyond  the  section 
limits;  and  when  the  contractor  has  removed  the  earth  from 
between  DI  and  Eit  indicate  to  him  those  exact  points  by 
marks  on  the  rock,  and  also  set  the  slope  stakes  at  D*  and  E*. 

235.  The  frequency  with  which  cross  sections  should 
be  taken  depends  entirely  upon  the  form  of  the  surface;  where 
this  is  regular,  a  section  at  each  station  is  sufficient.     A  cross 
section  should  be  taken,  not  only  at  every  point  on  the  centre 
line  where  there  is  an  angle  in  the  profile,  but  also  wherever 
an  angle  would  be  found  in  the  profile  of  a  line  joining  a  series 
of  slope  stakes  on  either  side,  even  though  the  profile  of  the 
centre  line  maybe  quite  regular  at  the  corresponding  point: — 
the  object  being,  not  only  to  indicate  the  proper  outlines  of 
the  earthwork,  but  to  furnish  the  data  necessary  to  calculate 
correctly  the  quantities  of  material  removed.     Rock  work  will 
generally  require  more  frequent  sections  than  earthwork. 

236.  Vertical  Curves. — The  grades  as  given  on  the 
profile  are  right  lines,  which  intersect  each  other  with  angles 
more  or  less  abrupt.     These  angles  require  to  be  replaced  by 
vertical  curves,  slightly  changing  the  grade  at  and  near  the 
point  of  intersection.     A  vertical  curve  rarely  need  extend 
more  than  200  feet  each  way  from  that  point.     Fig.  92. 


Let  AB,  SO,  be  two  grades  in  profile,  intersecting  at  station 
1?,  and  let  A  and  C  be  the  adjacent  stations.  It  is  required  to 
join  the  grades  by  a  vertical  curve  extending  from  A  to  C. 
Suppose  a  chord  drawn  from  A  to  C;—  the  elevation  of  the 
middle  point  of  the  chord  will  be  a  mean  of  the  elevations  of 
grade  at  A  and  C;  and  one  half  of  the  difference  between  this 


204  FIELD    ENGINEERING. 

and  the  elevation  of  grade  at  B  will  be  the  middle  ordinatc  of 
the  curve.     Hence  we  have 


in  which  M  —  the  correction  in  grade  for  the  point  B.  The 
correction  for  any  other  point  is  proportional  to  the  square  of 
its  distance  from  A  or  C.  Thus  the  correction  at  A  -\-  25  is 
^M  ;  at  A  -{-  50  it  is  \M  ;  at  ^l  -f-  75  it  is  ^M;  and  the  same 
for  corresponding  points  on  the  other  side  of  B.  The  correc- 
tions in  the  case  shown  are  subtractine,  since  M  is  negative. 
They  are  additive  when  M  is  positive,  and  the  curve  concave 
upward. 

These  corrections  are  made  at  the  time  the  cross  sections 
are  taken,  and  the  corrected  grades  are  entered  in  the  field- 
book  opposite  the  numbers  of  the  respective  stations. 

237.  Form  of  Field-book.—  A  complete  record  of 
all  cross-section  work  is  kept  in  the  cross-section  book. 
On  the  left-hand  page  is  recorded,  in  the  first  column,  the 
numbers  of  the  stations  and  other  points  where  sections 
are  taken  ;  in  the  second,  the  elevations  of  those  points,  copied 
in  part  from  the  location  level-book,  but  verified  or  corrected 
at  the  time  the  section  is  taken  ;  in  the  third,  the  elevation  of 
the  grade  for  the  same  points;  in  the  fourth,  the  width  of 
base  b  ;  in  the  fifth,  the  slope  ratios,  s  ;  and  in  the  sixth,  the 
surface  ratio  *'  when  uniform.  The  right-hand  page  has  a 
central  column,  in  which,  and  opposite  the  number  of  the 
station,  is  recorded  the  centre  depth  of  -the  section,  marked 
-f-  or  —  ,  to  indicate  cut  or  fill,  as  the  case  may  require. 
To  the  right  of  this  are  recorded  the  notes  of  that  portion  of 
the  section  which  lies  on  the  right  of  the  centre  line,  as  the 
line  was  run,  and  to  the  left,  the  notes  of  the  left  side.  The 
distance  from  the  centre  to  each  point  noted  is  recorded  as 
the  numerator  of  a  fraction,  and  the  cut  or  fill  at  the  point 
as  the  denominator,  prefixed  by  a  -f-  or  —  as  the  case  may 
require.  The  denominator  for  a  grade  point  is  zero.  The 
numbers  of  the  stations  should  increase  up  the  page,  as  in  a 
transit  book,  so  that  there  may  be  no  confusion  as  to  the  right 
and  left  side  of  the  line.  The  several  points  being  noted  in 
order  as  they  occur  from  the  centre  outwards,  the  notes  far- 


CONSTRUCTION. 


205 


thest  from  the  centre  of  the  page  usually  appertain  to  the 
slope  stakes ;  but  in  case  the  cross  profile  is  extended  beyond 
the  slope  stake,  the  note  of  the  latter  should  be  surrounded  by 
a  circle  to  distinguish  it.  The  following  form  is  a  specimen 
of  a  right-hand  page,  with  the  first  column  only  of  the  left- 
hand  page : 


Sta. 

Cross 

Sections. 

83 
+  60 
82 

+  38 
+  27 
+  19 

81 
80 

22.9      16.5       10 

5 
4 

o 

10 

20 
+25.6 
24 

32 

55.6 

+  8.6  +14  +17.7 
17.5       10 

+  21.5 
0 

o 

+  9^4 

o 

+20.8 
10 

+28.3 
'42.6 

+30.4 

+  5.0  +10 
14.2 

+13.2 
10 

+14.7 
6 

+20.1 
10  ' 

+21.7 
31.6 

+  2.8 
21.7 

+  5.4 
10 

+  8.5 
10 

+11.6 
19.3 

15 
-  5.3 

18 

+14.4 
25.6 

0 

7 

+  2.8 
0 
0 
-  4.7 
0 

+  3.8 
10 

-  9.8 
25.9 
-12.6 
33.4 

-  5.6 

7 

0 

7 

-11.2 

7 

-12 
0 

-17.6 

-16.4 

-17.6 

-19.6 

-19.1 

-12.4 

238.  In  case  there  is  a  liability  to  land-slips,  the  profiles 
of  cross  sections  should  be  carried  beyond  the  slope  stakes, 
on  the  upper  side  of  the  cut,  to  any  distance  thought  neces- 
sary to  reach  firm  ground,  and  stakes  driven  for  future  refer- 
ence.    When  a  number  of  consecutive  cross  profiles  are  to  be 
considerably  extended,  it  is  well  to  first  run,  instrumentally, 
a  line  parallel  to  the  centre  line,  and  set  stakes  opposite  the 
stations,  taking  their  elevations.     The  intermediate  surface  of 
the  sections  may  then  be  taken  with  cross-section  rods  if  more 
convenient.     See  §37.  . 

239.  In  case  of  inaccessible  ground,  preventing  a 
regular  staking  out,  an  indefinite  profile  of  the  section  may 
generally  be  obtained,  referred  to  the  datum  for  elevation  and 
to  the  centre  line  for  position,  which  being  plotted  on  cross- 
section  paper,  and  the  grade  Hue  and  side  slopes  added,  shows 
to  scale  where  the  slope  stakes  should  be. 


206  FIELD 


240.  Any  isolated  mass  of  rock  or  earth  which  oc- 
curs within  the  limits  of  the  slope  stakes,  but  not  included  in 
the  regular  notes,  is  separately  .measured  and  noted,  so  that 
its  contents  may  be  computed  and  added  to  the  sum  of  the 
same  material  found  in  the  cross  sections. 

241.  Borrow-pits.  —  When  the  excavations  will  not 
suffice  to  complete  the  embankments,  material  may  be  taken 
from   other   localities,  termed  borrow-pits.     These  should   be 
staked   out  by  the  engineer  and  their  contents   calculated, 
unless  the  contractor  is  to  be  paid  for  work  by  embankment 
measurements.     A  number  of  cross  profiles  are  taken  of  the 
original  surface,  and  (on  the  same  lines)  of  the  bottom  of  the 
pit  after  it  is  excavated,  which  furnish  the  depth  of  cutting 
at  each  required  point.     Borrow-pits  should  be  regularly  ex- 
cavated, so  that  they  may  not  present  an  unsightly  appear- 
ance   when  abandoned.      Borrow-pits   may  be  avoided  by 
widening  the  cut  uniformly  at  the  time  it  is  staked  out,  so 
that  it  may  furnish  sufficient  material;  provided  the  material 
is  suitable,  the  embankment  accessible,  and  the  distance  not 
too  great.     When  the  excavation  is  in  excess,  the  surplus  ma- 
terial should  be  uniformly  distributed  by  widening  the  adja- 
cent embankments,  if  possible;  otherwise  it  is  deposited  at' 
convenient  places  indicated  by  the  engineer  and  is  said  to  be 
wasted, 


242.  Shrinkage.  —  In  estimating  the  relative  amounts  of 
excavation  and  embankment  required,  allowance  must  be  made 
for  difference  in  the  spaces  occupied  by  the  material  before  ex- 
cavation and  after  it  is  settled  in  embankment.  The  various 
earths  will  be  more  compact  in  embankment,  rock  less  so.  The 
difference  in  volume  is  called  shrinkage  in  the  one  case,  and 
increase  in  the  other. 

Shrinkage  in  1000  cu.  yds. 
Material.  Of  excavation.    Of  settled  erubkt. 

Sand  and  gravel  ..............     80  C.  Yds.          87  C.  Yds. 

Clay  .........................  100     "  111     " 

Loam  ........................  120     "  136     " 

Wet  soil  .........  .  ...........  150     "  200     " 

Increase  in  1000  cu.  yds. 
Rock,  large  fragments  .........  600  C.  Yds.          375  C.  Yds. 

"      medium  fragments  ......  700     4<  413     " 

"      small  "  .  800     "  444    " 


CONSTRUCTION.  207 

Thus,  an  excavation  of  sand  and  gravel  measuring  1000  cubic 
yards  will  form  only  about  920  cubic  yards  of  embankment;  or 
an  embankment  of  1000  cubic  yards  will  require  1087  cubic 
yards  of  sand  or  gravel  measured  in  excavation  to  fill  it ;  but  will 
'require  only  587  cubic  yards  of  rock  excavation,  the  rock  being 
broken  into  medium-sized  fragments;  while  1000  cubic  yards 
of  the  latter,  measured  in  excavation,  will  form  1700  cubic 
yards  of  embankment. 

The  lineal  settlement  of  an  earth  embankment  will  be 
about  in  the  ratio  given  above,  therefore  the  contractor  should 
be  instructed  in  setting  his  poles  to  guide  him  as  to  the  height 
of  grade  on  an  earth  embankment,  to  add  10  per  cent  (average) 
to  the  fill  marked  on  the  stakes.  In  rock  embankments  this 
is  not  necessary.  The  engineer  should  see  that  all  embank- 
ments are  made  full  width  at  first,  out  to  the  slope  stakes,  and 
by  measure  at  or  above  grade,  so  that  the  whole  may  settle  in 
a  compact  mass.  Additions  to  the  width  made  subsequently 
are  likely  to  slide  off. 

243.  The  cross-section  notes  should  be  traced  in  ink  at  the 
first  opportunity  to  secure  their  permanence.     An  office  copy 
should  also  be  made  to  serve  in  case  of  loss  or  damage  to  the 
original. 

244.  Alteration  of  Line. — Inasmuch  as  the  centre  line 
at  grade  is  the  base  of  reference  for  all  measurements  and  cal- 
culations in  earthwork,  any  change  made  in  it  after  the  work 
of  grading  has  begun  should  be  most  carefully  recorded  and 
explained.     The  centre  stakes  of  the  old  line  should  be  left 
standing  until  after  the  new  line  is  established,  so  that  the  per- 
pendicular offset  from  the  old  line  to  the  new,  at  each  station, 
may  be  measured,  as  also  the  distance  that  the  new  station  may 
be  in  advance  of,  or  behind  the  old  one.   The  date  of  the  change 
should  be  recorded.     The  original  cross  sections  are  extended 
any  amount  requisite,  the  distance  out  being  still  reckoned  from 
the  old  centre,  while  a  marginal  note  states  the  amount  by  which 
the  centre  has  been  shifted. 

The  difference  in  length  of  the  lines  will  make  a  long  or  short 
station  at  the  point  of  closing.  The  exact  length  of  such  a 
station  should  be  recorded,  so  that  it  may  be  observed  in  re- 
tracing the  line  at  any  time,  and  in  calculating  the  quantity  of 


208  FIELD    ENGINEERING. 

earthwork.  The  original  transit  notes  of  the  altered  line  should 
be  preserved,  but  marked  as  "  abandoned,"  with  a  reference  to 
the  notes  of  the  new  line  on  another  page. 

245.  Drains  and  Culverts. — The  engineer  should  ex-' 
amine  the  nature  and  extent  of  each  depression  in  the  profile 
with  reference  to  the  kind  of  opening  required  for  the  passage 
of  water.  For  small  springs,  and  for  a  limited  surface  of  rain- 
fall, cement  pipes,  in  sizes  varying  from  12  to  24  inches  diame- 
ter, serve  an  excellent  purpose  as  drains.  These  are  easily  laid 
down,  and  if  properly  bedded,  with  the  earth  tamped  about 
them,  are  very  permanent ;  but  their  upper  surface  should  be 
at  least  2$  feet  below  grade.  The  embankment  is  protected  at 
the  upper  end  of  the  drain  by  a  bit  of  vertical  wall,  enclosing 
the  end  of  the  pipe.  If  necessary,  a  paved  gutter  may  lead  to 
it. 

Where  stone  abounds,  the  bed  of  a  dry  ravine  may  be  partly 
filled  with  loose  stone,  extending  beyond  the  slopes  a  few  feet, 
which  will  prevent  the  accumulation  of  water. 

When  the  flow  of  water  is  estimated  to  be  too  great  for  two 
lines  of  the  largest  cement  pipe,  or  when  the  embankment  is 
too  shallow  to  admit  them  safely,  a  culvert  is  required.  A 
pavement  is  laid  one  foot  thick,  protected  by  a  curb  of  stone 
or  wood  3  feet  deep  at  each  end,  and  wide  enough  to  allow  the 
Avails  to  be  built  upon  it.  It  should  have  a  uniform  slope,  usu- 
ally between  the  limits  of  50  to  1  and  100  to  1  to  ensure  the 
ready  flow  of  water.  In  firm  soils  the  foundation  pit  is  exca- 
vated one  foot  below  the  bed  of  the  stream,  but  if  mud  is  found 
this  must  be  removed  and  the  space  filled  with  riprap,  the  up- 
per course  of  which  is  arranged  to  form  the  pavement  at  the 
proper  level.  In  a  V-shaped  ravine,  requiring  too  much  ex- 
cavation at  the  sides,  and  where  the  fall  is  considerable,  riprap 
may  be  used  to  advantage,  the  bed  of  the  stream  above  the 
culvert  being  graded  up  by  the  same  material  to  meet  the  pave- 
ment. In  some  cases  a  curtain,  or  cross  wall,  is  necessary  on 
the  lower  end  to  retain  the  riprap. 

Culverts  should  be  laid  out  at  right  angles  to  the  centre  line 
whenever  practicable,  the  bed  of  the  stream  being  altered  if 
necessary.  The  length  of  an  open  culvert  is  the  entire,  distance 
between  slope  stakes,  the  walls  being  parallel  throughout,  or 
the  length  may  be  taken  somewhat  less  than  this,  and  the  walls 


CONSTRUCTION.  209 

turned  at  right  angles  on  the  upper  end,  forming  a  facing  to 
the  foot  of  the  slope.  The  walls  are  carried  up  to  grade  for 
the  width  of  the  road-bed,  and  are  stepped  down  to  suit  the 
slopes.  A  course  is  afterwards  added  to  retain  the  ballast. 

In  box  culverts  the  span  varies  from  2  to  5  feet,  the  height 
in  the  clear  from  2  to  6  feet;  the  thickness  of  walls  from  3  to 
4  feet;  the  thickness  of  cover  from  12  to  18  inches,  and  its 
length  at  least  2  feet  greater  than  the  span.  The  walls  terminate 
in  short  head-walls  built  parallel  to  the  centre  line,  the  top 
course  being  a  continuation  of  the  cover.  The  length  of  a 
head- wall,  measured  on  the  outer  face,  is  equal  to  the  height  of 
the  culvert  in  the  clear  multiplied  by  the  slope  ratio  of  the 
embankment.  The  perpendicular  distance  from  the  centre 
line  to  the  face  of  a  head-wall  is  equal  to  one  half  the  road-bed, 
plus  the  depth  of  the  top  of  the  wall  below  grade  multiplied  by 
the  slope  ratio,  or  ±b  -{-  sk.  A  coping  is  sometimes  added. 

24(>.  Arch  culverts  are  used  when  the  span  required  is 
more  than  5  feet,  and  the  embankment  too  high  to  warrant 
carrying  the  walls  up  to  grade  as  an  open  culvert.  The  span 
varies  from  6  to  20  feet;  the  arch  is  a  semicircle,  the  thickness 
varying  from  10  or  12  inches  to  18  or  20  inches.  The  height 
of  abutments  to  the  springing  line  varies  from  2  to  10  feet,  the 
thickness  at  the  springing  line  from  3  to  5  feet,  and  at  the  base 
from  3  to  6  feet,  the  back  of  the  abutment  receiving  the  batter. 
The  foundations  are  laid  broader  and  deeper  than  in  box  cul- 
verts, each  abutment  having  its  own  pit,  carried  to  any  depth 
found  necessary.  The  half  length  of  the  culvert  is  \b  -j-  sk,  in 
which  k  is  the  depth  of  the  crown  of  the  arch  below  grade. 
The  abutments  are  carried  up  half  way  from  the  spring  to  the 
level  of  the  crown  of  the  arch,  and  thence  sloped  off  toward 
the  crown.  The  face  walls  are  carried  up  to  the  crown,  and 
coped.  The  wing  walls  stand  at  an  angle  of  30C  with  the 
axis  of  the  culvert,  they  receive  a  batter  on  the  face,  and  are 
stepped  (or  sloped)  down  to  suit  the  embankment.  Their 
thickness,  at  the  base,  is  the  same  as  that  of  the  abutment;  at 
the  outer  end  3  feet.  They  stop  about  3  feet  short  of  the  foot 
ot  the  slope.  They  need  not  be  curved  in  plan. 

Any  stone  structure  of  dimensions  greater  than  those  given 
above,  scarcely  comes  under  the  head  of  culverts,  and  should 
be  made  the  subject  of  a  special  design  by  the  engineer. 


210 


FIELD 


247.  Staking-  out  Foundation  Pits.— For  box 
culverts. — The  engineer  having  decided  upon  the  size  of  cul- 
vert required,  makes  a  diagram  of  it  in  plan,  on  a  page  of  his 
masonry  book,  recording  all  the  dimensions,  stating  the  sta 
tion  and  plus  at  which  its  centre  is  taken,  the  span  and  height 
of  the  opening,  etc.  He  then  sets  the  transit  at  the  centre  A, 
Fig.  93,  measures  the  angle  between  the  centre  line  and  axis, 


Fio.  93. 

(making  it  90°  if  practicable) ;  on  the  axis  he  lays  off  the  dis- 
tances to  the  ends  of  the  culvert  and  drives  stakes  at  B  and  G. 
Perpendicular  to  BC  he  lays  off  the  half  widths  of  the  pit,  set- 
ting stakes  at  D  and  E,  and  laying  off  DFaud  EH  =  AB;  and 
DG  and  El  =  AC.  On  IG  produced  he  lays  off  CJ  =  OK,  and 
perpendicular  to  this  JM  and  KL,  and  finds  the  intersections 
0  and  N.  A  stake  is  driven  at  each  angle,  and  upon  it  is 
marked  the  cut  required  to  reach  the  assumed  level  for  the 
foundation.  These  cuts  are  recorded  on  the  corresponding 
angles  of  the  diagram.  The  pit  is  thus  no  larger  than  the 
plan  of  the  proposed  masonry,  and  the  sides  are  vertical,  which 
answers  the  purpose  for  shallow  pits. 

For  arch  culverts. — The  pit  for  each  abutment  when 
shallow  may  be  of  the  same  dimensions  as  the  lower  founda- 
tion course  .  if  more  than  five  feet  deep,  it  should  be  enlarged 
by  an  extra  space  of  one  foot  all  around.  In  Fig.  94  the  inside 


CONSTRUCTION.  211 

lines  show  the  plan  of  the  abutments  at  the  neat-lines ;  the 
outside  lines  represent  the  pits.  Having  prepared  a  plan  of 
the  structure  suited  to  the  locality,  and  made  a  diagram  of 
the  same  in  the  masonry  book,  set  the  transit  at  A,  and  drive 
stakes  at  D,  E,  N  and  0  on  the  centre  line.  Then  turning  to 
the  axis  BC,  lay  off  AC,  and  set  stakes  at  O  and  /.  With  Or 
as  a  centre,  and  a  radius  tqual  to  2DJS,  describe  on  the  ground 


>>c  
Fl 

B! 

I/i                           ^''  j" 

H: 
s<.  

y2        '' 

FIG.  94. 

an  arc  cutting  El  in  X  or  (IX =  DE .  cot  30°)  may  be  calcu- 
lated; and  on  XG  produced  lay  off  Q K,  and  perpendicular  to 
this,  KL.  From  N  lay  off  NP,  parallel  to  AC,  and  measure 
PL  as  a  check.  Drive  a  stake  at  each  angle,  marked  with  the 
proper  cutting,  and  record  the  same  on  the  diagram.  The 
locality  may  require  the  wings  to  be  of  different  lengths  and 
angles,  of  which  the  engineer  will  judge.  Guard-plugs  should 
be  driven  in  line  with  the  intended  face  of  one  or  both  abut- 
ments, so  that  the  neat-lines  can  be  readily  given  when  re- 
quired. In  case  the  material  is  not  likely  to  stand  vertically, 
the  pit  must  be  staked  out  with  sloping  sides,  as  described 
below.  -:'-* ':  ; 

For  bridge  abutments.— A  design  for  every  impor- 
tant structure  is  usually  prepared  in  the  office  after  a  survey 
of  the  site.  The  foundation  pit  is  then  laid  out  from  dimen- 
sions furnished  on  a  tracing,  but  a  diagram  of  the  pit  should  be 
made  in  the  masonry  book  as  usual.  When  the  bridge  is  on  a  tan- 
gent, Fig.  95,  set  the  transit  at  A  on  the  centre  line  at  its  inter- 
section with  the  axis  BC  oi  the  abutment ,  at  the  level  of  the  seat. 


212  FIELD  ENGINEERING. 

Deflect  from  the  tangent  the  angle  giving  the  direction  of  BC, 
and  lay  off  AC,  AB,  setting  plugs  at  B  and  C,  and  reference 
plugs  (two  on  each  side)  on  BC  produced.  After  staking  out 
the  sides  of  the  pit  parallel  to  BC,  set  the  transit  at  C,  and 
deflect  the  angle  for  the  wing,  laying  off  CD,  and  driving 
stakes  at  the  corners  E  and  F.  Two  reference  points  are 
then  set  on  the  line  CD  produced.  The  other  wing  being 


FIG.  95. 

staked  out  in  the  same  manner,  the  cut  is  found  at  each  stake 
and  marked  and  recorded.  Cross  sections  are  then  taken  near 
each  corner,  perpendicular  to  each  side,  and  slope  stakes 
(marked  "slope")  are  driven  where  the  slope  runs  out.  Inter- 
mediate sections  are  taken  when  the  unevenness  of  the  ground 
makes-  it  necessary,  and  the  lines  joining  the  slope  stakes  are 
produced  to  intersect,  and  other  stakes  are  driven  at  the  inter- 
sections. The  position  of  each  stake  is  shown  on  the  diagram, 
and  the  cut  recorded. 

A  slope  of  1  to  1  is  usually  sufficient  for  pits.  If  the  material 
will  not  stand  at  1^  to  1,  or  if  space  cannot  be  spared  for  the 
slope,  the  sides  may  be  carried  down  vertically,  supported  by 
sheet  piling  braced  from  within. 

The  reference  points  should  be  so  chosen  that  the  points  A, 
B  and  (7  may  be  found  by  intersection,  on  any  course  of  the 
masonry,  during  the  progress  of  construction. 

When  the  bridge  is  on  a  curve,  the  bridge-chord 
should  be  found  and  the  abutments  laid  out  from  this.  Fig.  96. 
The  bridge-chord  is  a  line  AB,  midway  between  the  chord  of 
the  curve  CD,  joining  the  centres  of  the  abutments,  and  a  tan- 
gent to  the  curve  at  the  middle  point  of  the  span.  Hence 


CONSTRUCTION. 


213 


CA  =  DB  =  $MNt  which  may  be  laid  off,  and  A  and  B  are 
the  true  centres  of  the  abutments,  from  which  the  foundations 
are  staked  out  as  before. 

The  distance  CE  =  DF  to  the  points  where  the  bridge-chord 
cuts  the  curve  is  0.147(71). 

Should  an  abutment  site  on  a  curve  be  inaccessible,  as  when 


FIG.  96. 

under  water,  from  any  transit  point  P  on  the  curve  lay  off.  PX 
perpendicular  to  the  tangent  at  M,  observing  that 

PX  —  MQ  —  A  C  =  R  (vers  PM—  \  vers  CM) 
and         A X  =  PQ-%AB  =  R  (sin  PM  - 


The  point  A  may  then  be  found  by  intersection,  or  by  direct 
measurement  with  a  steel  tape  or  wire,  driving  a  long  stout 
stake  to  show  the  point  above  the  water.  Other  points  may 
then  be  approximately  found,  sufficient  to  begin  operations. 

In  case  of  a  bridge  of  several  spans,  the  piers  are  laid  out  in 
the  same  manner,  from  a  centre  point  and  axis.  If  on  a  curve, 
each  span  has  its  own  bridge  chord,  but  for  convenience,  the 
centre  of  a  pier  may  be  taken  on  the  centre  line  during  its  con- 
struction, and  the  bridge-chord  only  found  for  the  purpose  of 
placing  the  bridge;  the  piers  being  long  enough  to  allow  of  the 
shift. 


214:  FIELD 

To  locate  the  centres  of  piers,  a  base  line  is  re 
quired  on  one  or  both  shores,  and  two  transits  are  used  to  give 
the  intersections  by  calculated  angles.  When  practicable  the 
spans  should  also  be  measured  with  a  steel  tape  or  wire. 

The  bed  of  a  pit  for  any  sort  of  structure  should 
receive  the  closest  scrutiny  of  the  engineer,  it  being  his  duty 
to  judge  whether  the  material  will  resist  the  load  to  be  im- 
posed upon  it.  A  pit  may  require  to  be  excavated  to  a  greater 
depth  than  first  ordered,  while  sometimes  a  less  depth  will 
answer,  as  when  solid  rock  is  found.  When  a  good  material 
is  reached,  if  any  doubt  exist  as  to  its  thickness,  or  as  to  the 
character  of  the  underlying  stratum,  borings  should  be  made 
or  sounding  rods  driven  down.  Piles  may  be  driven  to  gain 
the  requisite  firmness,  and  a  layer  of  riprap,  of  beton,  or  of 
timber  may  be  used  to  afford  a  uniform  bearing.  When  satis- 
fied of  the  stability  of  the  bed,  the  engineer  finds  the  original 
centres,  and  gives  points  for  the  courses  of  masonry.  A  com- 
plete record  is  kept  of  the  amount  and  kind  of  excavation,  the 
materials  used  in  foundation  under  the  masonry,  and  of  the 
size  and  thickness  of  each  foundation  course  of  masomy;  the 
notes  should  be  taken  at  the  time  the  work  is  done,  it  being 
generally  impossible  to  take  measurements  thereafter. 

248.  Cattle-guards  are  shallow  pita  placed  at  right 
angles  across  the  road  at  the  fence  lines  to  prevent  the  passage 
of  cattle.     They  are  either  entirely  open,  in  which  case  they 
should  be  at  least  4  feet  deep,  or  they  are  covered  in  part  with 
wooden  rails  laid  a  few  inches  apart.     The  open  guard  is 
preferred.     It  is  built  like  an  open  culvert  except  that  no 
pavement  is  required.     The  stringers  carrying  the  rails  over 
any  opening  should  be  no  longer  than  the  span  plus  the  thick- 
ness of  the  walls. 

249.  Trestle  Work. — No  wooden  culverts  should  ever 
be  used.     If  stone  cannot  be  had  at  first,  two  trestle  bents  may 
be  erected,  leaving  between  them  a  space  sufficient  to  contain 
the  stone  structure  to  be  built  when  the  material  for  it  can  be 
brought  by  rail.     The  bents  may  be  backed  by  plank  to  retain 
the  embankment,  and  the  stringers  are  then  notched  down  an 
inch  on  the  caps  to  receive  the  pressure  of  the  earth,  and 
render  the  bents  mutually  sustaining.     The  sills  are  prevented 
from  yielding  to  the  pressure  cf  the  earth  by  being  sunk  in 


COKSTBUCTIOK.  215 

a  trench,  or  by  sheet  piling.  Should  the  span  be  too  long,  a 
central  bent  may  be  used,  so  as  not  to  interfere  with  building 
the  wall.  Sometimes  pile-bents  may  be  used  with  greater  ad- 
vantage, the  piles  being  driven  in  rows  of  four  each,  and  cap- 
ped to  receive  the  stringers.  In  districts  where  suitable  stone 
is  entirely  wanting,  pile  or  trestle  abutments  and  piers  are 
used  for  the  support  of  bridges,  the  piles  or  posts  being 
arranged  in  groups  and  capped  to  receive  the  direct  weight  of 
the  trusses.  They  should  not  sustain  the  embankment,  but 
should  be  connected  with  it  by  a  short  trestle  work. 

Trestle  work  is  frequently  used  as  a  substitute  for  embank- 
ment, either  to  lessen  the  first  cost,  or  to  hasten  the  completion 
of  the  line,  or  for  lack  of  suitable  material  with  which  to  form 
an  embankment.  The  cost  of  trestle  work,  however,  is  not 
less  than  that  of  an  earth  embankment  formed  from  borrow 
pits,  unless  its  height  exceeds  about  15  feet,  depending  on  the 
relative  prices  of  materials  and  labor.  When  not  exceeding  30 
feet  in  height,  the  bents,  for  single  track,  are  usually  composed 
of  two  posts,  a  cap  and  sill,  each  12  X  12,  and  two  batter  posts, 
10  X  12,  inclined  at  ±th  to  1,  all  framed  together.  Two  lengths 
of  3-inch  plank  are  spiked  on  diagonally  on  opposite  sides  of 
the  bent  as  braces.  The  length  of  the  caps  should  equal  the 
width  of  the  embankment;  the  posts  should  be  5  feet  from 
centre  to  centre,  ai^d  the  batter  posts  2  feet  from  the  posts  at 
the  cap.  The  sill  should  extend  about  two  feet  beyond  the 
foot  of  the  batter  post.  A  masonry  foundation  for  the  bent  is 
preferable,  though  pile  foundations  are  not  uncommon,  and 
some  temporary  structures  are  placed  directly  on  a  firm  soil, 
supported  only  by  mudsills  laid  crosswise  under  the  sill.  The 
spans,  or  distance  between  bents,  may  vary  from  12  to  16  feet. 
The  stringers  should  consist  of  4  pieces,  2  under  each  rail, 
bolted  together,  with  packing  blocks  to  separate  them  2  or  3 
inches.  Over  each  bent  and  at  the  centre  of  each  span  a  piece 
of  thick  plank  about  4  feet  long  should  be  placed  on  edge 
between  the  two  pair  of  beams  to  preserve  the  proper  distance 
between  them,  while  rods  pass  through  the  beams  and  strain 
them  up  to  the  ends  of  the  plank,  to  increase  the  stability  of  the 
beams  and  prevent  their  buckling  under  a  load.  The  string- 
ers should  be  able  to  carry  safely  the  heaviest  load  without 
bracing  against  the  posts.  The  bents,  however,  if  high,  must 
be  braced  against  each  other.  The  stringers  should  be  con- 


216  FIELD 


linuous,  the  two  pieces  breaking  joints  with  each  other  at  the 
bents,  to  which  they  are  firmly  bolted.  They  may  rest  directly 
on  the  caps,  or  corbels  may  intervene.  The  spans  on  a  curve 
should  be  shorter  than  on  a  tangent.  The  ties  should  be 
notched  down  to  fit  the  stringers  closely,  and  guard  rails,  cither 
wood  or  iron,  secured  to  them  firmly.  Unless  the  spans  are 
very  short,  horizontal  bracing  should  be  employed  consisting 
of  3-inch  plank,  extending  from  the  centre  ol  each  span  to  the 
ends  of  the  caps,  which  are  notched  down  to  receive  the  plank. 

For  trestles  much  higher  than  30  feet  the  cluster  l>ent  is 
preferable,  so  termed  because  each  vertical  post  is  composed  of 
a  cluster  of  four  pieces,  8x8,  standing  a  little  apart  to  allow 
the  horizontal  members  to  pass  between  them.  The  verticals 
are  continuous,  breaking  joints,  two  and  two,  while  the  hori- 
zontals pass  the  posts  and  are  bolted  to  them  at  the  joints;  the 
framing  is  accomplished  entirely  by  packing  blocks  and  bolts. 
The  batter  posts  consist  each  of  two  pieces  8x8;  the  horizon- 
tals may  be  4  X  10,  and  extend  not  only  across  the  bent,  but 
from  one  bent  to  another.  Proper  bracing  is  also  used  in  every 
direction.  When  very  high,  a  secondary  pair  of  batter  posts 
may  be  introduced  in  the  lower  part  of  the  structure.  The 
batter  need  not  exceed  £tli  to  1.  In  some  instances  two  adjoin- 
ing bents  are  strongly  braced  together,  forming  a  tower  or  pier, 
and  the  piers  placed  from  50  to  100  feet  apart,  the  roadway 
being  carried  on  trussed  bridges.  The  cluster  bent  admits  of 
any  piece  being  removed  and  a  new  one  inserted  when  neces- 
sary. 

Iron  trestles  are  now  adopted  where  a  permanent  struc- 
ture is  desired.  Owing  to  the  expansion  of  the  metal  by  heat, 
the  bents  cannot  be  continuously  connected  with  each  other  as 
in  a  wooden  trestle;  hence  the  pier  form  is  resorted  to,  having 
spans  varying  from  30  to  150  feet,  covered  by  trussed  bridges, 
and  the  whole  structure  is  more  properly  styled  a  viaduct. 

2oO.  Tunnels.  Tunnels  are  adopted  in  certain  cases  to 
avoid  excessive  excavations,  steep  grades,  high  summits,  and 
circuitous  routes.  Their  disadvantages  are  the  increased  time 
and  cost  of  their  construction  compared  with  an  open  line,  and 
their  lack  of  light  and  fresh  air  when  in  use.  It  is  desirable 
that  they  should  be  on  a  tangent  throughout,  both  for  the  ad- 
mission of  light  and  for  convenience  of  alignment.  Many 


,   CONSTRUCTION".  217 

tunnels,  however,  liave  been  built  with  a  curve  at  one  or  both 
ends.* 

The  location  of  a  tunnel,  other  things  being  equal,  should 
be  such  as  to  make  not  only  the  tunnel  proper,  but  also  its  im- 
mediate approaches  by  open  cut  as  short  as  possible ;  and  the 
latter  should  be  selected  so  as  not  to  be  subject  to  overflow, 
nor  liable  to  land  slides.  The  material  to  be  encountered  may 
frequently  be  determined  with  tolerable  accuracy  by  a  study 
of  the  geological  formation  in  the  vicinity,  or  by  actual  borings. 
The  most  favorable  material  for  tunnelling  is  a  homogeneous 
self-supporting  rock,  devoid  of  springs,  which  does  not  disin- 
tegrate on  exposure  to  the  atmosphere.  The  worst  materials 
are  saturated  earth  and  quicksands.  The  presence  of  water  in 
any  material  increases  the  cost  considerably. 

The  alignment  of  a  tunnel  is  made  the  subject  of  special 
survey,  after  the  general  location  is  decided,  and  this  is  more 
or  less  elaborate  according  to  the  length  of  tunnel.  A  perma- 
nent station  is  established  at  the  highest  point  crossed  by  the 
tunnel  tangent,  from  which,  if  possible,  monuments  are  set  in 
each  direction  at  points  beyond  the  ends  of  the  tunnel.  If 
there  are  two  principal  summits,  stations  on  these  will  define 
the  tangent,  which  may  then  be  produced.  The  monuments 
established  beyond  the  tunnel  should  be  sufficiently  distant  to 
afford. a  perfect  backsight  from  the  ends  of  the  tunnel,  where 
other  monuments  arc  also  established.  The  first  quality  of  in- 
struments only  should  be  used,  and  these  perfectly  adjusted, 
and  the  observations  should  be  repeated  many  times  until  it  is 
certain  that  all  perceptible  errors  are  eliminated.  Since  the 
line  of  collimation  will  be  frequently  inclined  to  the  horizon 
at  a  considerable  angle,  it  is  important  that  it  should  revolve 
in  a  vertical  plane;  and  to  secure  this,  a  sensitive  bubble  tube 
should  be  attached  to  the  horizontal  axis,  at  right  angles  to  the 
telescope  of  the  transit.  The  distance  may  be  obtained  by  tri- 
angulation,  though  direct  measurement  is  to  be  preferred.  A 
steel  tape  is  convenient  and  accurate,  providing  that  allowance 
be  made  for  variations  due  to  temperature,  from  an  assumed 
standard.  The  rods  described  in  §  43  may  be  used  instead  of 


*  The  Mont  Cenis  tunnel,  requiring  a  curve  at  each  end,  was  first 
opened  on  the  tangent  produced,  giving  a  straight  line  through,  and  the 
curves  were  excavated  subsequently. 


218  FIELD   ENGINEERING. 

plumb  lines,  the  tape  being  held  at  right  angles  to  them,  and 
therefore  horizontal.  A  plug  should  be  driven  for  each  rod  to 
stand  on,  and  a  centre  set  to  indicate  the  line  and  measure- 
ment. 

As  the  excavation  of  the  tunnel  proceeds,  the  centre  line  is 
given  at  short  intervals  by  points  either  on  the  floor  or  roof. 
Overhead  points  are  generally  preferred,  from  which  short 
plumb  lines  may  be  hung,  constantly  indicating  the  line,  with 
little  danger  of  being  disturbed.  When  a  new  transit  point  is 
required  in  the  tunnel,  it  should  be  established  directty  under 
an  overhead  point,  which  serves  as  a  check  upon  its  perma- 
nence, and  as  a  backsight  when  needed. 

Shafts  are  sometimes  opened  to  give  access  to  several  points 
of  the  tunnel  at  the  same  time,  thus  facilitating  the  work,  though 
at  an  increased  cost.  They  also  serve  for  ventilation  during  the 
progress  of  the  work,  though  they  are  worse  than  useless  for 
this  purpose  afterward,  except  possibly  in  the  case  of  a  single 
shaft  near  the  centre  of  the  tunnel.  Some  of  the  longest  tun- 
nels have  been  formed  without  shafts,  while  many  shorter  ones 
have  had  several,  which  have  generally  been  closed  after  the 
tunnel  was  completed.  Shafts  are  either  vertical,  inclined,  or 
nearly  horizontal ;  in  the  latter  case  they  are  called  adits.  In- 
clined shafts  should  make  an  angle  of  at  least  60°  with  the  ver- 
tical. Vertical  shafts  may  be  either  rectangular,  round,  or 
oval.  Their  dimensions  vary,  depending  on  their  depth  and 
the  material  encountered,  between  8  and  25  feet.  They  are 
usually  sunk  on  the  centre  line  of  the  tunnel,  though  some- 
times at  one  side.  When  over  the  tunnel  the  alignment  below 
is  obtained  directly  from  two  plumb  lines  of  fine  wire  suspended 
on  opposite  sides  of  the  shaft  from  points  very  carefully  deter- 
mined at  the  surface.  The  plummets  are  suspended  in  water 
to  lessen  their  vibrations,  and  as  soon  as  the  transit  can  be  set 
up  at  a  sufficient  distance  to  bring  the  lines  into  focus,  it  is 
shifted  by  trial  into  exact  line  with  the  mean  of  their  oscilla- 
tions, the  latter  being  very  limited.  Permanent  points  may 
then  be  set,  but  should  be  repeatedly  verified.  As  soon  as  the 
workings  from  a  shaft  communicate  with  those  from  either 
end,  or  from  another  shaft,  the  alignment  thus  found  is 
tested,  and  revised  if  necessary.  These  operations  require  the 
greatest  nicety  of  observation  and  delicacy  of  manipulation  to 
obtain  satisfactory  results. 


CONSTRUCTION.  219 

From  plumb  lines  in  the  central  shaft  of  the  Hoosac  tunnel, 
the  line  was  produced  three  tenths  of  a  mile,  and  met  the  line 
produced  2.1  miles  from  the  west  end  with  an  error  in  offset 
of  five  sixteenths  of  an  inch.  In  the  Mont  Cenis  tunnel  the 
lines  met  from  opposite  ends  with  "  no  appreciable"  error  in 
alignment,  while  the  error  in  measurement  was  about  45  feet 
in  a  total  length  of  7.6  miles. 

When  a  curve  occurs  in  a  tunnel  it  is  usually  near  one 
end.  The  tunnel  tangent  is  produced  and  established  as 
before  described,  and  a  second  tangent  from  some  point  on  the 
curve  outside  the  tunnel  is  produced  to  intersect  it,  the  inter- 
section being  precisely  determined  and  the  angle  measured 
with  many  repetitions.  The  tangent  distances  are  then  calcu- 
lated, and  the  position  of  the  tangent  points  corrected  by 
precise  measurements,  and  permanent  monuments  are  estab- 
lished. As  the  tunnel  advances,  points  may  be  set  at  short 
intervals  on  the  curve  in  the  usual  manner;  but  at  intervals 
of  100  feet  the  regular  stations  should  be  defined  with  finely 
centred  monuments,  using  a  100-foot  steel  tape  carefully  sup- 
ported in  a  horizontal  position.  When  it  is  necessary  to  use  a 
subchord,  its  exact  length  should  be  calculated  as  shown  in 
§  107.  When  the  curve  has  advanced  so  far  as  to  render  a  new 
transit  point  necessary,  this  should  be  established  at  a  full 
station.  The  subtangents  from  the  two  transit  points  should 
then  be  produced  to  intersect,  and  measured  for  equality  with 
each  other  and  with  their  calculated  length.  The  distance 
from  their  intersection  to  the  middle  of  the  long  chord  should 
also  be  measured  as  a  check  on  the  deflections.  When  no 
perceptible  errors  remain,  the  curve  may  be  produced  as 
before  until  the  P.  T.  is  reached.  It  is  evident  that  correct 
measure  is  indispensable  to  correct  alignment  on  curves. 
Should  obstacles  on  the  surface  necessitate  triangiilation,  more 
than  ordinary  care  must  be  exercised,  and  as  many  checks 
introduced  as  possible.  The  triangles  should  be  so  arranged 
that  all  of  the  angles  and  most  of  the  sides  may'  be  measured. 

Test  levels  are  carried  over  the  surface  with  great  care, 
each  turning  point  being  made  a  permanent  bench,  and  its 
elevation  determined  with  a  probable  error  not  exceeding 
0.005  foot.  Levels  may  be  carried  down  a  shaft  on  a  series  of 
bolts  or  spikes  about  12  feet  apart  in  the  same  v.ertical  line, 
the  distances  being  measured  by  the  same  level-rod  as  that 


220  FIELD    ENGINEERING. 

with  which  the  benches  are  determined.  The  measures  should 
be  taken  between  two  graduations  of  the  rod,  not  using  the 
end  of  the  rod,  wThich  may  be  slightly  worn.  Fine  horizontal 
lines  on  the  heads  of  the  bolts  may  be  used  to  mark  the  exact 
distances.  After  the  shaft  reaches  the  level  of  the  tunnel,  the 
depth  may  be  measured  more  directly  with  a  steel  tape,  the 
entire  length  of  which  has  been  corrected  at  the  given  tem- 
perature, by  comparison  with  the  same  rod. 

If  the  grade  of  a  tunnel  is  to  be  continuous,  it  should 
be  assumed  at  something  less  than  the  maximum  of  the  road, 
but  not  less  than  0.10  per  station,  which  is  required  for 
drainage.  If  a  summit  is  to  be  made  in  the  tunnel,  the  grade 
from  the  upper  end  should  not  exceed  0.10  per  station. 
Grades  are  given  in  the  tunnel  from  day  to  day,  or  as  often  as 
required  by  the  progress  of  the  work,  the  marks  being  made 
on  the  sides  at  some  arbitrary  distance  above  grade.  Turning 
points  should  be  taken  on  permanent  benches? 

The  least  width  of  a  tunnel  in  the  clear  should  be,  for 
single  track  about  15  feet,  and  for  double  track  26  feet.  The 
least  height  in  the  clear  above  the  tie  should  be  18.5  feet 
for  single  track,  and  16.5  feet  at  the  outside  rails  for  double 
track,  allowing  for  tie  and  ballast;  the  roof  at  the  centre  of  the 
section  should  be  at  least  20  feet  above  subgrade,  and  with  a 
full  centred  arch  22  or  23  feet  for  double  track.  The  form 
of  section  depends  somewhat  on  the  material  traversed.  In 
perfectly  solid  rock  a  nearly  rectangular  section  may  be  used, 
the  roof  being  slightly  rounded.  In  dry  clay,  and  stratified 
rock,  a  flat  arch  may  be  used,  and  in  other  cases  a  full-centred 
arch.  The  latter  form  is  rather  to  be  preferred  on  account  of 
the  better  ventilation  afforded.  The  sides  are  made  vertical, 
battered  or  curved,  as  necessity  or  taste  may  dictate.  In  wet 
and  infirm  soil  an  invert  floor  may  be  required,  otherwise  it 
is  made  level  transversely.  When  a  lining  is  required  the 
original  section  must  of  course  be  made  large  enough  to 
allow  for  the  masonry,  and  the  temporary  timber  supports 
behind  it.  Hard  burned  brick  is  usually  adopted  for  arching, 
being  durable  and  easily  handled.  In  loose  rock  the  arching 
may  be  from  13  to  26  inches  thick,  in  wet  and  yielding  soil  a 
thickness  of  from  2G  to  39  inches  may  be  necessary.  The 
walls  may  be  from  2J  to  6  feet  thick. 

In  forming  a  tunnel,  a  heading   or  gallery  of  smaller 


CONSTRUCTION.  221 

cross  section  is  first  driven  and  afterwards  enlarged  to  the 
full  size  required.  In  firm  clay  or  loose  rock  which  will  tem- 
porarily support  itself  until  the  masonry  can  be  put  in,  it  is 
better  to  drive  the  heading  along  the  floor  (at  subgrade)  of  the 
tunnel,  the  remaining  material  being  then  easily  thrown  down 
in  sections  as  the  arching  is  advanced.  In  solid  rock,  or  wet 
earth,  a  top-heading  (along  the  roof)  is  generally  preferred. 
The  dimensions  of  a  heading  driven  by  hand  are  usually  8  feet 
high  by  8  or  10  feet  wide,  but  in  solid  rock  where  drilling 
machinery  is  introduced,  it  is  advantageous  to  make  the  head- 
ing as  wide  as  the  tunnel  at  once.  By  drilling  holes  into  the 
face  at  points  about  five  feet  each  side  of  the  centre,  and  con- 
verging on  the  centre  line  at  a  depth  of  about  ten  feet,  a  tri- 
angular mass  of  rock  may  be  blown  out,  and  the  space  thus 
gained  facilitates  the  blasting  of  the  adjacent  rock  on  either 
side.  An  advance  of  about  10  feet  in  each  day  of  24  working 
hours  may  thus  be  made,  using  nitroglycerine  in  some  form 
as  the  explosive  agent.  Owing,  however,  to  unavoidable 
delays  from  various  causes,  this  rate  of  progress  cannot 
always  be  maintained.  At  the  Hoosac  tunnel  the  greatest 
advance  in  one  week  was  50  feet;  in  one  month  184  feet  at 
one  heading.  At  the  Musconetcong  tunnel  a  heading  8  X  22 
feet  in  syenitic  gneiss  was  advanced  at  the  average  rate  of 
137  feet  per  month  for  6  months,  the  maximum  being  144  feet 
— the  enlargement  of  the  tunnel  to  full  size  going  on  at  the 
same  time,  a  few  hundred  feet  behind.  At  the  St.  Gothard 
tunnel  the  north  heading  2. 5  X  3  metres  was  advanced  in 
mica  gneiss,  during  the  year  1875  at  the  average  daily  rate  of 
3.71  metres,  with  a  maximum  of  about  4  metres,  but  the  en- 
largement was  not  made.  The  south  heading  advanced  at 
the  rate  of  2  metres  a  day,  timbering  being  at  times  necessary. 

In  ordinary  clay  a  heading  may  be  driven  at  from  75  to  180 
ft.  per  month,  according  to  circumstances,  where  timbering  is 
put  in.  The  enlargement,  including  timbering  and  masonry, 
may  be  advanced  at  from  20  to  60  ft.  per  month.'  Small  tun- 
nels for  water  conduits  are  driven  through  dry  clay  at  the  rate 
of  10  ft.  per  day,  the  masonry  following  at  once  without  tim- 
bering. 

The  compressed  air  used  to  drive  the  drilling  machinery 
serves  to  supply  ventilation  also.  When  this  is  wanting  or 
proves  insufficient,  exhaust  fans  are  used.  At  Mont  Cenis  a 


FIELD    ENGINEERING. 

horizontal  bi'attice  or  partition  was  built  in  the  tunnel,  dividing 
it  so  as  to  secure  a  circulation  of  air.  When  foul  gases  are  en- 
countered, ventilation  becomes  a  serious  question,  and  in  one 
instance  an  important  work  was  abandoned  for  this  cause. 

Cross  sections  of  the  heading,  and  also  of  the  tunnel  en- 
largement, should  be  measured  at  intervals  of  about  20  feet,  as 
soon  as  opened,  to  see  that  the  sides,  roof,  and  floor  are  taken 
out  to  the  prescribed  lines,  at  the  same  time  that  the  latter  are 
exceeded  as  little  as  possible.  In  solid  rock,  since  some  ma- 
terial outside  of  the  true  section  will  necessarily  be  thrown 
down,  leaving  an  irregular  outline,  it  is  well  to  take  two  cross 
sections  at  the  same  point,  one  following  the  projections  and 
the  other  the  recesses  of  the  rock,  from  which  an  average  sec- 
tion may  be  estimated.  A  daily,  or  at  least  a  weekly,  record 
of  operations  should  be  kept  in  tabular  form,  and  the  progress 
indicated  by  a  profile  and  cross  sections  drawn  on  a  sufficiently 
large  scale  to  show  details. 

The  drainage  of  a  tunnel  is  best  secured  by  a  line  of 
stoneware  or  cement  pipe  laid  in  a  trench  along  each  side,  and 
covered  with  ballast  or  other  loose  material.  The  entire  floor 
is  thus  made  available  for  the  use  of  the  trackmen.  When  an 
invert  is  used,  the  drain  is  placed  in  the  centre  between  tracks. 
If  the  amount  of  water  is  large,  drain  pipe  may  be  laid  behind 
the  walls,  and  the  back  of  the  arch  may  be  covered  with  as- 
phaltum,  or  coal  tar,  to  prevent  a  constant  dripping  on  the 
track. 


Retracing  the  Line.—  As  the  grading  pro- 
gresses, in  either  excavation  or  embankment,  the  principal 
transit  points  are  established  on  the  road-bed  from  the  points 
of  reference,  and  the  centre  line  is  retraced,  setting  stakes  at 
every  50  feet.  Transit  points  on  grade  should  be  fixed  upon 
stout,  durable  posts  firmly  set  in  the  ground,  and  standing 
high  enough  to  be  easily  reached  after  the  ballast  is  laid. 
To  recover  the  old  line,  any  discrepancies  in  measurement 
must  be  left  between  the  transit  points  where  they  occur, 
and  not  carried  forward.  In  retracing  a  curve,  if  the  transit 
is  placed  at  the  forward  point,  allowing  the  chain  to  ad- 
vance toward  it,  slight  differences  in  measurement  will  not 
affect  the  position  of  the  curve.  If  any  short  or  long  sta- 


CONSTRUCTION.  223 

* 

tions  have  been  introduced  on  the  location,  their  position  on 
the  line  must  not  be  changed  in  retracing.  The  chain  may 
be  adjusted  so  that  its  measures  will  agree  with  the  recorded 
distances  between  transit  points.  Offsets  are  made  right  and 
left  from  the  new  stakes  to  see  that  the  road-bed  is  of  the  full 
width  at  all  points.  The  levels  are  also  .carried  over  the 
grade,  and  any  remaining  cut  or  fill  found  necessary  is  marked 
on  the  back  of  the  stakes,  due  allowance  being  made  for  the 
probable  settlement  of  embankments. 

252.  As  the  work  approaches  completion  the  contractor 
goes  over  the  line  dressing  it  to  grade  and  opening  .the  side 
ditches  if  this  has  not  been  previously  done. 

Drain-tile  should  be  laid  at  the  bottom  of  these  ditches  and 
lightly  covered  with  earth,  particularly  if  the  cut  be  wet. 
These  not  only  prevent  the  water  from  reaching  the  ballast, 
but  by  keeping  the  foot  of  the  slope  comparatively  dry  pre- 
vent the  earth  from  sliding  down  and  filling  up  the  cut. 
There  is  also  a  marked  economy  in  their  use,  as  the  cost  is 
trifling,  and  all  further  excavation  of  mud  and  water  from 
the  cut  is  generally  obviated.  Should  any  springs  appear  in 
the  slope  a  branch  line  of  smaller  tile  may  be  laid  to  meet  it. 
If  the  slope  is  liable  to  be  overflowed  from  the  surface  above, 
an  open  ditch  should  be  dug  a  few  feet  beyond  the  slope 
stakes,  leading  the  surface  water  to  discharge  elsewhere. 

253.  The  road-bed  being  prepared,  ballast  stakes  are 

driven  at  every  half  station,  giving  the  width  of  the  ballast  at 
its  base,  while  the  tops  of  the  stakes  indicate  the  proper  level 
of  its  upper  surface,  which  is  the  under  side  of  the  tie.  These 
stakes  should  be  set  so  as  to  give  the  proper  elevation  to  the 
outer  rails  on  curves  when  the  ballast  is  graded  to  them.  The 
ballast  should  be  about  one  foot  deep  before  the  ties  are  laid. 
Broken  stone  or  a  mixture  of  coarse  and  fine  gravel  is  the 
best  material,  affording  elasticity  and  good  drainage.  The 
side  slopes  of  the  ballast  are  made  1  to  1 ;  its  width  at  the 
under  side  of  the  tie  should  be  one  foot  greater  than  the 
length  of  the  tie. 

254.  Track-laying. — After  the  ballast  has  been  laid 
and  graded,  the  centre  line  is  retraced  upon  it ;  short  stakes 


224  FIELD   ENGINEERING. 

i 

are  used,  each  of  which  is  centred.  On  long  tangents,  one 
stake  in  every  200  feet  is  sufficient,  on  ordinary  curves  one  in 
every  50  feet,  and  on  very  sharp  curves  one  In  every  25  feet. 
The  ties  are  then  spaced  evenly  according  to  the  number 
prescribed  per  mile,  or  per  rail  length ;  but  a  tie  should  not  be 
allowed  to  cover  a  transit  point.  Ties  for  the  standard  gauge 
are  8  or  9 feet  long;  they  should  be  sawed  off  square  at  the  ends 
and  in  uniform  lengths  for  appearance  sake  when  laid. 
Specifications  usually  call  for  ties  having  a  thickness  of  6 
inches  and  a  width  of  from  7  to  10  inches.  The  ends  of  the 
ties  are  aligned  on  one  side  of  the  road,  though  if  cut  into 
uniform  lengths  both  ends  will  be  equally  well  aligned.  The 
rails  are  then  laid  on,  and  spiked  to  gauge.  The  first  spikes 
are  driven  in  the  ties  near  a  centre  stake,  the  centre  mark  of 
the  gauge  bar  being  kept  over  the  centre  on  the  stake.  Upon 
curves  the  rails  must  be  sprung  to  the  proper  arc  before  they 
are  laid  (§  199).  All  the  ties  required  in  a  given  distance 
should  be  laid  before  the  rails  are  brought  upon  them.  The 
practice  of  laying  only  joint  and  middle  ties  at  first  subjects 
the  rails  to  the  danger  of  bending  from  passing  loads. 

Owing  to  the  expansion  of  the  rails  by  heat,  a  space 
must  be  left  at  the  rail-joints.  The  highest  temperature  of  a 
rail  in  the  summer  sun  is  about  130°  Fah.  The  expansion  of 
iron  or  steel  per  100°  is  .0007  per  foot;  or  for  a  30-foot  rail 
.021  foot  or  .252  inch.  Therefore  when  30-foot  rails  are  laid 
at  a  temperature  near  the  freezing  point,  or  100°  below  the 
maximum,  the  space  allowed  must  be  at  least  a  quarter  of  an 
inch.  At  80°  Fah.  or  50°  below  the  maximum,  it  need  be  only 
half  as  much.  The  space  required  is  also  proportional  to  the 
length  of  rail  used.  The  exact  space  should  be  given,  as  less 
would  result  in  the  rails  being  forced  up  by  expansion,  while 
more  than  necessary  space  gives  a  rough  road,  and  hastens 
the  destruction  of  the  rail. 

Wherever  siding's  are  required,  the  necessary  frogs  and 
long  switch- ties  should  be  provided  in  advance,  so  that  they 
may  be  put  in  place  at  the  time  of  laying  the  main  track.  For 
every  road  crossing  at  grade,  heavy  oak  plank  should  be  pro- 
vided, and  laid  upon  the  ties  as  soon  as  the  rails  are  spiked, 
so  that  the  highway  travel  may  not  be  impeded. 


CALCULATION   OF   EARTHWORK.  225 


CHAPTER  X. 
CALCULATION  OF  EARTHWORK. 

254.  The  first  step  toward  finding  the  cubical  content  of 
an  excavation  is  to  divide  it  into  a  number  of  prismoids  by 
several  cross  sections. 

A  prismoid  is  a  solid  having  plane  parallel  bases  or  ends, 
and  bounded  on  the  sides  either  by  planes,  or  by  such  surfaces 
as  may  be  generated  by  a  right  line  moving  continuously  along 
the  edges  of  the  bases  as  directrices. 

The  positions  of  the  cross  sections  must  be  so  selected 
that  the  solid  included  between  any  two  consecutive  sections 
may  be  a  prismoid  as  nearly  as  possible.  Upon  a  tangent  the 
roadbed  and  side  slopes  are  planes,  so  that  the  prismoidal 
character  of  a  given  solid  depends  upon  the  shape  of  the  natu- 
ral surface.  When  the  natural  surface  is  a  plane,  the  sections 
are  taken  only  at  the  regular  stations,  100  feet  apart;  when  it 
is  curved,  warped,  irregular,  or  broken,  the  sections  must  be 
more  numerous,  so  that  the  surface  limited  by  any  two  shall 
be  composed  substantially  of  right-lined  elements  extending 
from  one  section  to  the  other. 

If  two  end  sections  of  a  prismoid  are  somewhat  similar,  we 
infer  that  the  corresponding  points  are  connected  by  right- 
lined  elements,  forming  in  each  case  the  axis  of  a  ridge  or  of  a 
hollow.  If  one  section  has  less  breaks  than  the  next,  some  of 
these  ridges  or  hollows  must  vanish;  and  in  order  that  the 
solid  may  be  a  prismoid,  they  must  vanish  in  the  section  of 
least  breaks ;  therefore  a  cross  section  must  be  taken  on  the 
ground  through  the  point  where  each  ridge  or  hollow  vanishes, 
and  the  distance  of  that  point  from  the  centre  line  noted,  so 
that  it  may  be  coupled  with  the  proper  point  in  the  next  section 
for  exact  calculation  of  content. 

When  ridges  or  hollows  run  diagonally  across  the  line  of 
roa'd,  cross  sections  must  be  taken  where  they  are  intersected 
not  only  by  the  centre  line  but  also  by  the  side  slopes;  that  is, 
sections  must  be  taken  so  that  a  side  stake  may  stand  on  top  of 


226  FIELD    ENGINEERING. 

each  ridge  and  at  bottom  of  each  hollow.  In  case  the  centre 
line  intersects  at  right  angles  a  retaining  wall  or  other  vertical 
surface,  two  cross  sections  are  required  at  the  same  point,  one 
at  top  and  the  other  at  base  of  wall,  in  order  to  furnish  the 
data  necessary  to  calculate  the  content  each  way  from  the  ver- 
tical surface.  (See  Art.  235.) 

Every  thorough  cut  terminates  in  either  side -hill  cutting,  a 
pyramid,  or  a  wedge;  the  latter  happens  only  when  the  con- 
tour of  the  natural  surface  is  at  right  angles  to  the  line  of  road. 
Sections  should  always  be  taken  through  the  points  where  the 
edges  of  the  road  bed  meet  the  surface,  as  these  are  the  points 
of  separation  between  thorough  and  side  hill  work.  Such  sec- 
tions also  serve  to  define  terminal  pyramids  when  they  occur 
as  is  illustrated  by  Fig.  97.  In  side-hill  work  the  foregoing 


Fio.  97. 

rules  apply  as  well,  but  sections  will  generally  be  more  numer- 
ous than  in  thorough  cuts,  The  same  rules  apply  also  to  em- 
bankment, but  as  grading  is  preferably  paid  for  in  excavation, 
the  same  precision  in  determining  the  quantities  in  embank- 
ment is  not  usually  necessary. 


CALCULATION    OF   EARTHWORK.  227 

255.  Formulae  for  Sectional  Areas. 

Let  b  =  base  of  section  or  width  of  road-bed, 

horizontal 

s  =  slope  ratio  =  —  —  :  —  i— 
vertical 

"  d  =  depth  at  centre  stake. 

"   h,  k  =  depths  at  side  stakes. 

"  m,  n—  horizontal  distances  from  centre  to  side  stakes. 

For  ground  level  transversely,  the  section  is  a  parallelogram, 
and  the  area  is  evidently 


(342) 
or  directly  from  the  field  notes, 

A  =  $(b  +  m  +  ri)d  (343) 

For  ground  of  uniform  transverse  slope  between  slope  stakes, 


Fig.  98,  the  section  consists  of  the  parallelogram  ABOE  and 
the  triangle  EOD.     Hence 

A  =  &AB+  EO)EN+  ±EO(DH-EN) 
A  -  \(AB  .  EN+  EO  .  DH) 


or 

A  =  i[ 

also  (344) 

A  =  $[bk+7i(b-{-28k-)]} 

From  which  also 

A  —  tyh  4-  mk  ) 
and  [  (345) 


These  formulae  are  independent  of  the  centre  depth.     They 
are  convenient  for  calculating  the  area  of  a  plotted  section 


228 


FIELD    ENGINEERING. 


having  an  irregular  surface  after  the  surface  line  has  been 
averaged  by  stretching  a  silk  thread  over  it.  The  points 
where  the  thread  intersects  the  slope  lines  determine  the 
values  of  h,  k,  m,  and  n  respectively. 

When  the  ground  has   uniform  slopes  transversely  from  the 
centre  to  the  side  stakes:  Fig.  99 :  If  in  the  diagram  we  draw 

D 


!r 


FIG.  99. 

EG  and  DG,  the  section  will  be  divided  into  four  triangles, 
two  having  the  common  base  CG  —  d  and  respective  heights 
GN=  m  and  GH=  n,  and  two  having  the  equal  bases  AG  — 
GB  =  $  and  the  respective  heights  EN  =  h  and  DH  =  A;. 
Hence  we  have  for  the  area  of  section 


A  = 


(346) 


Othencise,  if  the  slope  lines  are  produced  to  meet  below 
grade  at   P,  then    GP  — =  --.     The  area  of  CEPD  is 

8  ,i/j 


NII=  %(d  +  |J  (m  +  7i).    The  area  of  ABP  is  J.  G 
GP  —  -7-     Hence  we  have  for  the  area  of  the  section 


X 


(347) 


Both  these  formulas  are  convenient,  and  as  the  values  of  the 
several  letters  can  be  substituted  directly  from  the  field  notes, 
it  is  unnecessary  to  plot  such  sections. 

.  When  the. surface  of  the  ground  is  irregular,  verticals  are  con- 
ceived to  be  drawn  to  the  grade  line  through  the  slope  stakes, 


CALCULATION   OF   EARTHWORK.  229 

and  through  each  break  in  the  surface  line,  giving  a  number  of 
trapezoids,  the  areas  of  which  are  severally  calculated,  and 
from  their  sum  is  subtracted  the  area  of  the  two  triangles  EN  A 
and  DUE.  The  remainder  is  the  area  of  section  required. 
This  calculation  maybe  made  directly  from  the  data  furnished 
by  the  field  notes  without  plotting;  but  if  the  ground  has  a 
number  of  small  breaks,  it  is  generally  better  to  plot  the  sec- 
tions and  stretch  an  averaging  line  over  them,  finding  the  areas 
by  eq.  (345).  Or  two  averaging  lines  may  be  employed  extend- 
ing from  the  centre  stake,  each  way,  when  the  area  may  be  cal- 
culated by  cq.  (34G)  or  (347). 

256.  Prismoidal  Formulae  for  Solid  Contents. 

—  The  content  of  a  prismoid  may  be  exactly  calculated  by 
means  of  the  Prismoidal  Formula,  which  is 

8  =  [A  +  **+*}  (348) 


8  =  cubic  yards,  I  =  length  in  feet,  A,  A'  —  the  areas  at  the 
two  parallel  ends,  and  M  —  the  area  of  a  section  midway  be- 
tween the  ends.  This  area  is  not  a  mean  of  the  other  two,  but 
the  linear  dimensions  of  the  mid-section  are  means  of  the  cor- 
responding dimensions  severally  of  the  end  sections;  from 
which  therefore  the  area  of  the  mid  -section  may  be  computed. 
The  labor  of  calculating  the  middle  area  may  be  avoided  in 
many  instances  by  substituting  in  the  prismoidal  formula,  eq. 
(348),  for  A,  A,  and  M,  their  values  as  given  in  eq.  (342)  for 
ground  level  transversely. 

„..     td  4-  d!   .    d*-\-2dd'-{-d'* 
A=bd-}-sd*    A'  =  bd'  -f-  sd'*    M=b~^  --  h*--1-  —  7— 


8  = 


in  which  S  is  expressed  in  terms  of  the  end  dimensions. 

257.  Tables  of  cubic  yards  may  be  constructed  upon  this 
formula  which  are  very  convenient  in  practice.  The  constant 
values  in  any  one  table  are  I  which  is  taken  at  100,  and  b  and  s 
which  are  given  values  corresponding  to  the  road-bed  and  slope 
ratio.  The  variables  are  d  and  d'.  The  columns  in  the  table 


230  FIELD   ENGINEERING. 

will  be  headed  by  the  successive  values  of  d'  ,  while  each  hori- 
zontal line  will  be  headed  by  a  value  of  d.  For  any  one 
column  therefore  d'  is  constant,  and  the  only  variable  is  d. 
Assuming  any  value  for  d',  the  values  of  8  in  that  column  may 
be  computed,  letting  d  take  a  series  of  values  differing  by  unity 
from  zero  upwards,  and  the  corresponding  values  of  S  will  be 
placed  in  the  column  d'  opposite  the  several  values  of  d. 

But  instead  of  solving  the  eq.  (349)  for  each  value  of  S  re- 
quired, the  process  of  rilling  the  table  may  be  much  abbre- 
viated by  observing  that  since  the  equation  is  of  the  second 
degree  with  respect  to  the  variable  d,  the  second  difference  of 
the  values  of  S  will  be  a  constant  and  equal  to  twice  the  co- 

efficient of  $t  or  d"  =       8         Also  the  first  term  in  the  series 
D  X  <*  7 

of  first  differences  of  8  in  the  column  d'  (i.e.  between  d  =  0 
and  d  =  1)  is  expressed  by  the  sum  of  the  coefficients  of  d* 
and  d't  or 


The  first  value  of  S  in  any  column  d  '  is  found  by  solving 
eq.  (349)  after  making  d  =  0;  or, 


Starting  with  these  values  we  may  fill  any  column  d  '  simply 
by  successive  additions.  The  values  of  d'  for  the  several 
columns  should  also  differ  by  unity.  The  final  value  of  S  in 
each  column  should  be  calculated  by  formula  as  a  check;  or 
since  all  the  final  quantities  in  the  same  line  d  of  the  table 
form  a  series  of  which  the  second  difference  is  d",  if  on  taking 
their  differences  this  result  is  obtained,  the  quantities  are 
proved  to  be  correct. 

Example.  —  Given  a  base  of  18  feet  and  slopes  1-J  to  1,  to  fill 
the  column  of  d'  =  6  in  a  table  of  cubic  yards  for  level  cross 
sections.  Here  I  =  100,  b  =  18,  s  =  f,  d'  —  6.  Hence  d'  = 
3.7037+,  <V  =  46.2962-J-,  and  £„  =  266.6666-J-.  It  is  not 
necessary  to  go  beyond  the  fourth  decimal  place,  since  that 
figure  will  always  be  the  sain'e  as  the  first  decimal  (a  result 


CALCULATION   OF   EARTHWORK.  231 


due  to  dividing  by  27),  and  may  be  corrected  by  it  after  every 

addition. 

The  process  is  as  follows: 

.  'V 

d 

o                                                   o 

6" 

0 

266.6666                       A«  OQ«O 

1 

312  9629                      tto.^yo/v 

3.7037 

2 

362.'9629 

3.7037 

3 

4 

416.6666 
474.0740                    Zi  irn 

3.7037 
3.7037 

5 

385.1851                      Si'ftiift 

3.7037 

6 

600.0000                      AftSiS 

3.7037 

7 

668.5185                      799900 

3.7037 

8 

740.7407                     75'.9259 

3.7037 

etc.                             etc. 

etc. 

In  copying  into  the  table,  the  quantities  are  taken  to  the 
nearest  unit  only,  and  the  decimals  are  otherwise  neglected. 

The  completed  table  furnishes  values  of  8  corresponding  to 
any  values  of  d  and  d '  in  even  feet.  The  correction  for  the 
decimal  parts  of  the  depths,  when  there  are  such,  is  made  by 
adding  to  8  (found  opposite  the  even  feet)  the  product  of  the 
half  sum  of  the  decimals  by  the  difference  between  8  as 
found  and  the  next  value  of  8  diagonally  below  to  the  right. 
These  differences  may  for  convenience  be  inserted  originally 
in  the  table  under  each  quantity  in  small  figures.  •*, 

If  the  length  of  the  solid  differs  from  100  feet,  multiply  the 
corrected  quantity  by  the  length  and  divide  by  100,  since  S 
varies  directly  as  L  Such  tables  are  published  in  separate 
sheets  for  a  variety  of  bases  and  slopes,  so  that  usually  one 
may  be  purchased  to  suit  the  case  in  hand. 

258.  These  tables  may  be  used  to  find  quantities  when 
the  ground  is  not  level  transversely,  by  finding,  first,  the  area 
of  the  actual  sections,  and  second,  the  depths  of  level  sec- 
tions having  equal  areas,  and  then  using  the  depths  so  found 
as  the  values  of  d  and  d'  in  the  table  of  quantities.  The 
depths  of  equivalent  level  sections  are'called  equiva- 
lent depths.  They  may  be  calculated  by  the  formula 

&    ,     ./A.P 


which  is  derived  directly  from  ieq.  (342).  The  more  convenient 
method,  however,  is  to  construct  a  table  on  eq.  (342),  giving  to 
d  a  series  of  values  varying  by  one  tenth  of  a  foot  from  zero 


232  FIELD   ENGINEERING. 

upward.    The  values  of  b  and  s  in  this  table  must  agree  with 
those  of  the  road^and  of  the  table  of  cubic  yards. 

259.  When  the  transverse  slope  is  uniform  between  slope 
stakes  the  equivalent  depth  may  be  expressed  in 
terms  of  the  centre  depth  and  slope  of  surface 

without  reference  to  the  area,    Fig.  100. 

D 


FIG.  100. 
Let  EABD  be  the  given  section. 

"  RABT  be  the  equivalent  level  section. 
Produce  the  side  slopes  to  meet  at  P,  and  let  c  =  CP  and 

Through  C  draw  the  horizontal  line  QL,  and  at  L  erect  the 
perpendicular  LM  —  z,  and  draw  LN  parallel  to  PE. 

The  area  RPT  =  area  EPD;  and  QEC  =  LNG;  hence  area 
EPLN=  QPL  —  EPD  -  NLD. 

Since  NLD  is  similar  to  EPD,  we  have 

EPD  :  NLD  ::  c2 .-  z» 

or  EPD  -  NLD  :  EPD  ::  c2  -  z*  :  c* 

Since  QPL  and  MPT  are  similar, 

QPL  :  RPT  ::  c2  :  d2 


(7L        cs  c2  s2 

Let »'  =  slope  ratio  of  surface  =    ,,,-  =  — .  Then 22  =  — 75- 

JU.L      z  s 

which  substituted  gives 


CALCULATION   OF   EARTHWORK.  233 

If  di  —  the  equivalent  depth  (A  G,  then 

l-  c) 


A.  table  may  be  prepared  giving  (GI  —  c)  for  various  inclina- 
tions of  surface  with  given  base  and  side  slopes.  It  is  then 
only  necessary  to  add  this  correction  to  the  centre  depth  to 
obtain  the  equivalent  depth.  Such  tables  of  correction  usually 
accompany  the  published  tables  of  cubic  yards.  This  method 
of  obtaining  quantities  is  particularly  applicable  to  preliminary 
estimates,  where  the  ground  has  not  been  cross-sectioned,  and 
only  the  centre  depth  and  transverse  inclination  is  known. 

260.  The  use  of  the  earthwork  tables  described  gives 
correct  results  ;— 

1st.  When  the  surface  of  the  prismoid  is  a  plane,  however 
much  inclined;  provided  it  does  not  intersect  the  road-bed 
within  the  limits  of  the  prismoid. 

2d.  When  with  regular,  or  three-level  end  sections,  generally 
similar  to  each  other,  the  surface  is  regularly  warped  from  one 
end  to  the  other;  provided  that  the  side  lines  and  centre  lirie 
of  the  surface  are  straight,  and  that  no  two  of  them  are  in- 
clined to  grade  in  opposite  directions. 

3d.  When  the  ridges  or  hollows  of  an  undulating  surface 
are  parallel  to  the  line  of  the  road. 

4th.  When  a  surface  of  numerous  irregularities  may  be 
averaged  by  planes  or  warped  surfaces  so  as  to  comply  with 
one  of  the  preceding  conditions. 

But  the  method  fails  on  undulating  ground  when  the 
ridges  or  hollows  run  obliquely  to  the  line  of  road,  even 
though  the  sections  may  appear  quite  regular. 

In  general,  the  method  of  equivalent  depths  holds  good 
when  the  mid-section  of  the  equivalent  level  end  sections 
equals  in  area  the  actual  mid-section  or  the  prismoid;  other- 
wise it  fails. 

261.  The  content  of  a  prismoid  may  be  approximately  ob- 
tained by  the  method  of   mean    areas,  the  formula  for 
which  is 

(353) 


Although  approximate,  this  method  is  much  employed  on 


234  FIELD   EKGI^EERIKG. 

account  of  its  convenience.     It  is  approved  by  statute  to  be 
used  upon  the  public  works  of  the  State  of  New  York. 

If  the  values  of  A  and  A1  derived  from  eq.  (342).  be  substi- 
tuted in  eq.  (352),  and  then  eq.  (349)  be  subtracted  from  it  there 
remains 


which  is  the  correction  by  which  S  obtained  by  eq.  (352)  must 
be  diminished  to  make  it  equal  to  8  obtained  by  eq.  (349) 
when  the  ground  is  level  transversely. 

Again,  for  three-level  sections,  if  the  values  of  A,  A',  and  M 
derived  from  eq.  (347)  be  substituted  in  both  eq.  (348)  and 
(352),  and  one  subtracted  from  the  other,  there  remains 

[(d  -d'}  (m  +  n  -(m'  +  n'}  )  ] 


which  is  the  correction  by  which  S  obtained  by  eq  .  (352)  must 
be  diminished  to  make  it  equal  to  S  obtained  by  eq.  (349). 
Hence  we  may  write  at  once,  for  three  level-sections,  the  cor- 
rect formula: 


(353) 


This  formula  gives  results  identical  with  eq.  (349),  is  applica- 
ble to  the  same  cases,  and  gives  correct  results  or  fails  to  do 
so  according  to  the  conditions  stated  in  the  previous  section. 

262.  When  the  conditions  of  the  surface  are  such  that  eq. 
(349)  or  eq.  (353)  will  not  give  correct  results,  the  area  of 
the  mid-section  may  be  derived  from  its  calculated  linear 
dimensions  as  stated  in  §  256.  The  contents  of  the  prismoid 
are  then  given  .by  eq.  (348). 

Example  1.  (Fig.  101.)— Base  20.     Slopes  1$  :  1. 


A  =  200  sq.  ft. 


CALCULATION   OF   EAKTHWOKK. 


235 


If  I  =  100,  eq.  (348) 
100 


2°0)  =  "3  C' 


Had  this  been  calculated  by  eq.  (349)  or  eq.  (353)  or  by  the 


Fia.  101. 

tables,  the  result  would  be  1167  c.  yds.,  showing  an  error  of 
174  cubic  yards  in  excess. 


Example  2.  (Fig.  102.)— Base  20.     Slopes  H  >  1. 

22      ,     0  J.9^ 

6 


234  sq.  ft. 

88  sq.  ft. 

164. 5  sq.ft. 


If  I  =  100,  eq.  (348) 
100 


658 


-  605  c> 


Had  this  been  calculated  by  eq.  (349)  or  eq.  (353)  or  by  the 


236 


FIELD 


tables,  the  result  would  be  584  c.  yds.,  showing  an  error  of 
21  cubic  yards  in  deficit. 

263.  At  the  termination  of  a  cut  or  fill  we  have  usually 
either  a  wedge  or  a  pyramid.  To  a  wedge  the  pre- 
ceding formulae  and  tables  based  on  them  apply  by  making 


.    FIG.  102. 

one  end  depth  equal  zero.  In  the  case  of  a  pyramid,  the 
content  is  equal  to  the  area  of  the  section  forming  the  base 
multiplied  by  one  third  the  length  of  the  solid,  and  divided  by 
27  or 


264.  Side-hill  Work.—  When  the  natural  surface  has 
a  regular  transverse  slope  and  intersects  the  road-bed,  the 
cross  section  is  reduced  to  a  triangle.  If  w  =  the  intercepted 
portion  of  the  road-bed,  and  k  =  the  side  height,  then  A  = 
%wk.  Similarly  A'  -  %w'k'  and'4Jf  =  i(w  -f  w')  (k  -f  k'\ 
which  substituted  in  eq.  (348)  give 


8  = 


12X27 


5»  [(**  + 


p)*']         (355) 


which  is  convenient  for  direct  calculation  from  the  field  notes. 
It  is  not  adapted  to  the  construction  of  tables,  since  it  contains 
four  independent  variables. 


CALCULATION   OF   EARTHWORK.  237 

If  the  slope  of  the  natural  surface  is  given,  let  s'  be  the  sur- 
face slope  ratio  at  one  section,  and  s"  that  at  the  other,  and  s 
the  ratio  of  the  side  slope.  Then  w  =  k  (s'  —  s)  and  w'  = 
k'(s"  —  s),  which  substituted  in  eq.  (355)  give 


Jo  ^= 


6X27| 

If  the  surface  is  a  plane,  then  s"  =  *',  and  we  have  for  this 
case 

(356) 


which  is  a  formula  of  quite  limited  application  ;  yet  it  is  the 
one  on  which  tables  and  diagrams  are  usually  constructed. 
Consequently  the  latter  will  not  give  correct  results,  except 
when  the  surface  is  a  plane. 

265.  When  the  natural  surface  is  broken  the 

sections  may  be  plotted,  and  the  values  of  w  and  k  taken  from 
the  points  where  ail  averaging  line  intersects  the  grade 
and  side  slope  respectively.  Finding  values  for  w'  and  k'  in 
the  same  way,  the  content  may  then  be  obtained  by  eq.  (355) 
as  before.  The  averaging  line  should  not  only  cut  off  the 
same  area  as  the  original  section,  but  should  also  have  in  each 
case  a  slope  agreeing  as  nearly  as  possible  with  the  general 
slope  of  the  natural  surface.  The  slope  is  determined  simply 
by  inspection  of  the  diagram,  but  the  area  may  be  bad  pre- 
cisely, for,  taking  w  from  the  averaging  line,  and  knowing  A, 

O  A 

we  may  calculate  k  by  the  formula  k  =  —  ;    or  k  may  be 

taken  from  the  plot  and  w  calculated. 

Otherwise,  the  actual  mid-section  may  be  calculated  and 
the  cubic  contents  determined  by  the  method,  illustrated  in 
§262. 

266.  To  express  side-hill  areas  and  cubic  yards 
in  terms  of  the  centre  depth,  d,  and  transverse 
slope-ratio  s'.    Fig.  103. 


238  FIELD  ENGINEERING. 

For  any  depth  d,  add  to  this  area 


and  there  results, 


A=- 


K&  +  s'df 

2  X  27(«'  -s)  J 


(357) 


Observe  that  d  may  be  plus  or  minus,  and  that  its  limits  are 


Tables  of  cubic  yards  may  be  constructed  on  this  formula, 
making  d  and  s'  the  variables,  which  would  be  extremely  con- 


FIG.  103. 

venient  for  making  up  estimates  upon  preliminary  lines  on 
which  the  profile  of  centre  line  and  angle  of  transverse  slope 
only  are  known.  Since  s'  is  the  cotangent  of  the  slope  angle 
the  columns  of  the  table  may  be  headed  by  the  angles  in  a 
series  of  degrees,  while  the  corresponding  values  of  s'  are 
used  in  the  formula.  The  values  of  d  should  vary  by  tenths 
of  a  foot.  The  results  obtained  by  eq.  (356)  and  eq.  (357)  will 
be  identical  for  the  same  sections. 

267.  Several  different  systems  of  diagrams  have  been 
devised  and  published  for  determining  quantities  in  earthwork 
by  a  sort  of  graphical  method.  These  diagrams,  which  are 
substitutes  for  tables  are  preferred  by  some  engineers.  They 


CALCULATION   OF   EABTHWORK.  239 

are  based  on  the  same  principles,  and  are  constructed  on  modi- 
fications of  the  same  formulae. 


268.  Correction  of  Earthwork  for  Curvature. 

— The  preceding  calculations  are  based  on  the  assumption  that 
the  centre  line  is  straight,  with  cross  sections  at  right  angles  to 
it.  When  an  excavation  is  on  a  curve,  the  cross  sections,  be- 
ing in  radial  planes,  are  inclined  to  each  other,  so  that  the  con- 
dition of  a  prismoid  is  not  exactly  fulfilled.  But  by  the  proper- 
ty of  Guldinus,  if  any  plane  area  is  made  to  revolve  about  an 
axis  in  the  same  plane,  the  volume  of  a  solid  generated  by  the 
area  is  equal  to  that  of  a  prism  having  a  base  equal  to  the  given 
area,  and  a  height  equal  to  the  length  of  path  described  by  the 
centre  of  gravity  of  the  area.  The  path,  being  the  arc  of  a  cir- 
cle, is  proportional  to  the  radius  drawn  to  the  centre  of  gravi- 
ty. If  therefore  a  cross  section  is  symmetrical  with  respect  to 
the  centre  line,  the  path  of  the  centre  of  gravity  is  equal  to  the 
measured  length  of  the  centre  line,  and  no  correction  for  cur- 
vature is  required. 

But  when  the  ground  is  inclined  transversely,  the  centre  of 
gravity  is  one  side  of  the  centre  line,  and  its  path,  if  we  con- 
ceive it  to  sweep  around  the  curve,  from  one  end  of  a  prismoid 
to  the  other,  is  longer  or  shorter  than  the  distance  measured  on 
the  centre  line,  according  as  the  centre  of  gravity  is  outside  or 
inside  of  the  centre  line  curve. 

Let  C  =  correction  in  cubic  yards  due  to  curvature. 
"    S  =  cubic  yards  as  obtained  by  prismoidal  formula. 
"  R  =  radius  of  centreline. 
"    e  =  eccentricity  of  centre  of  gravity  of  section, 

=  horizontal  distance  from  centre  line  to  centre  of 
gravity. 

"We  then  have  the  proportion,  •;,.. 

8±C  :  8::  E±e  :  R 

-I 

As  the  sections  of  a  solid  are  seldom  similar  and  equal,  we 
shall  usually  have  a  different  value  of  e  for  every  section,  from 


240 


FIELD    ENGINEERING. 


which,  however,  a  mean  average  value  may  be  deduced,  and 
used  in  the  above  formula.  But  it  will  be  more  convenient  to 
correct  the  areas  themselves  for  eccentricity  before  finding  8, 
which  will  then  require  no  correction.  For  the  same  result 

will  ensue  whether  we  multiply  S  by  -^-,  or  multiply  one  of 
the  component  factors  of  8  by  the  same  ratio. 

If  then  c  =  correction  of  area  in  square  feet  due  to  eccentri- 
city, we  have  at  once 

Ae 


and  the  corrected  area  equals  A  ±  c  according  as  the  cut  is 
deeper  on  the  outside  or  inside  of  the  curve.  Each  area  used 
in  determining  the  solid  contents  should,  on  a  curve,  be  first 
corrected  in  this  manner. 

To  find  the  value  of  e  for  any  three-level  section.     Fig.  104. 


! _ 


FIG.  104. 


Find  the  areas  either  side  of  the  centre  line  separately,  call- 
ing them  .ff  and  JT,  and  take  their  sum  and  difference.  Using 
the  same  notation  as  in  §  255,  H  —  $md  -f  $bh,  K  =  %nd  -f- 
&k,  and  II  -f  K  =  A. 


-  H  = 


(n  -  m)  -f  &  (k  -  h) 


In  the  figure  draw  CE'  equal  to  CE,  and  the  triangle  CE'D 
will  represent  the  area  (K  —  H}.  Bisect  the  side  E'D,  and 
draw  a  line  from  C  to  the  middle  point.  Then  the  centre  of 


CALCULATION   OF   EARTHWORK.  241 

gravity  of  the  triangle  will  be  on  this  line  at  two  thirds  its 
length  from  C,  and  the  horizontal  distance  of  the  centre  of 
gravity  from  C  is  £  X  l(m  -f-  n)  —  i(w  +  n).  The  centre  of 
gravity  of  the  remainder  of  the  section  is  on  the  centre  line 
CO,  so  that  the  value  of  e  is  found  from  the  proportion 

K-  H-.A 


n-\-m 


Hence 


-m}  +  &  <*- 


(358) 


Sections  which  are  more  irregular  may  be  plotted  and 
reduced  by  averaging  lines  to  three-level  sections,  in  order  that 
the  formula  may  be  applied.  If  the  ground  is  so  irregular  as 
to  require  the  computation  of  the  middle  section,  the  correc- 
tion c  should  be  found  and  applied  to  this  area  (M )  also 
before  introducing  it  into  the  prismoidal  formula.  As  the 
correction  for  curvature  is  always  relatively  small,  it  is  usually 
ignored  in  practice  for  thorough  cuts,  except  where  deep  cut- 
tings with  stcsp  transverse  slope  occur  on  sharp  curves. 

The  correction  is  of  more  importance  relatively  in 
side-hill  work:  as  the  centre  of  gravity  of  the  section  is 
more  remote  from  the  centre  line.  Let  the  section  be  reduced 


FIG.  105. 

to  a  triangle  by  an  averaging  line  (Fig.  105),  and  w  be  the  base 
of  the  triangle  formed  by  the  averaging  line.  The  centre  of 
gravity  is  at  one  third  the  horizontal  distance  from  the  middle 
point  of  w  to  the  side  stake  D,  while  the  distance  of  this 
middle  point  from  the  centre  stake  C  is  evidently  %b  —  %w. 


242  FIELD   ENGINEERING. 

Hence  e  =  $b  —  \w  +  \\n  -  (#>  -  |w)] 

or  e  =  #b  -\-  n  —  w) 

Ae        b  -4-  n  —  w       wk 

and  C  =  -R~-    ^  —  XTT  .     (3o9> 

The  correction  c  will  be  plus  or  minus  as  before  explained. 
This  formula  applies  to  all  side-hill  triangular  sections, 
whether  there  be  cut  or  fill  at  the  centre  stake. 

Example  1.  —  Thorough  cut;  base  20;  slopes  H  •  1- 
I  =  100;  8°  curve,  left;  R  =  716.78 


4'  +--  +       -      _ 

2   "      0   "h  20 
Then  K  =  i  X  58  X  12  -f  i  X  20  X  32  =  508 

J5T=i  X  16  X  12  +  i  X  20  X    4  =  116.\  4  =  624 
K-  H—  392 


(4  +  c)   637.49 

JT;  =  |  x  40  X  8  +  i  X  20  X  20  =  260 
H'  =  i  X  13  X  8  +  i  X  20  X  2  =  62  .  •.  A'  =  322 

K-H=  198 

J3  +  40_  _ 

~  3  X  716.78  l 


=  326.87 


From  which  we  obtain  $  =  1758  cub.  yds.  —  Ans. 
Without  correction  we  have  1726     "      " 


Showing  a  difference  of  32     "      " 

Had  the  curve  been  to  the  right  with  same  notes,  c  would 
have  been  minus,  and  8  would  =  1694. 


CALCULATION   OF   EARTHWORK.  243 

Example  2.  —  Side-hill  cut;  base  20;  slopes  li  :  1 
I  =  60;  10°  curve,  right;  E  =  573.69 
6          0         40 


_0__L_2__,    37 
0.8^0.0"^  "18 
A  =  i  X  16  X20  =  160 


(J.  -  e)  =  156.42 

=  $  x  8  X  18  =  72 


(A!  -c)=    69.95 

Hence  8  =  248  cub.  yds. 

Without  correction  S  would  =  255     "       " 


Difference          7 

269.  Haul. — The  cost  of  removing  excavated  material, 
when  the  distance  does  not  exceed  a  certain  specified  limit,  is 
included  in  the  price  per  cubic  yard  of  the  material  as  meas- 
ured in  the  cutting.  But  when  the  material  must  be  carried 
beyond  this  limit,  the  extra  distance  is  paid  for  at  a  stipulated 
price  per  cubic  yard,  per  100  feet.  The  extra  distance  is  known 
by  the  name  of  haul,  and  is  to  be  computed  by  the  engineer 
with  respect  to  so  much  of  the  material  as  is  affected  by  it. 

The  contractor  is  entitled  to  the  benefit  of  all  short  hauls 
(less  than  the  specified  limit),  and  material  so  moved  should  not 
be  averaged  against  that  which  is  carried  beyond  the  limit. 
Therefore,  in  all  cuts,  the  material  of  which  is  all  deposited 
within  the  limiting  distance,  no  calculation  of  liaul  is  to  be 
made. 

On  the  other  hand,  the  company  is  entitled,  in  cases  of  long 
haul,  to  free  transportation  for  that  portion  of  the  cutting,  no 
one  yard  of  which  is  carried  beyond  the  specified  limit.  There- 
fore, this  portion  is  first  to  be  determined  in  respect  to  its  ex- 
tent; and  the  number  of  cubic  yards  contained  in  it  is  to  be  de- 


244  FIELD   ENGINEERING. 

ducted  from  the  total  content  of  the  cutting,  before  estimating 
the  haul  upon  the  remainder.  Find  on  the  profile  of  the  line 
two  points,  one  in  excavation,  and  the  other  in  embankment, 
such,  that  while  the  distance  between  them  equals  the  specified 
limit,  the  included  quantities  of  excavation  and  embankment 
shall  just  balance.  These  points  are  easily  found  by  trial,  with 
the  .aid  of  the  cross  sections  and  calculated  quantities,  and  be- 
come the  starting  points  from  which  the  haul  of  the  remainder 
of  the  material  is  to  be  estimated. 


FIG.  106. 

Pig.  106  represents  a  cut  and  fill  in  profile.  The  distance 
AB  is  the  limit  of  free  haul.  The  materials  taken  from  AO 
just  make  the  fill  OB  and  without  charge  for  haul;  but  the  haul 
of  every  cubic  yard  taken  from  AC,  and  carried  to  the  fill  BD, 
is  subject  to  charge  for  the  distance  it  is  carried,  less  AB.  It 
would  be  impossible  to  find  the  distance  that  each  separate  yard 
is  carried,  but  we  know  from  mechanics  that  the  average  dis- 
tance for  the  entire  number  of  yards  is  the  distance  between 
the  centres  of  gravity  of  the  cut  AC,  and  of  the  fill  BD  which 
is  made  from  it.  If,  therefore,  X  and  T  represent  the  centres 
of  gravity,  the  actual  average  haul  is  the  sum  of  the  distances 
(AX-\-BY),  and  this  (expressed  in  stations)  multiplied  by  the 
number  of  cubic  yards  in  the  cut  AC,  gives  the  product  to 
which  the  price  for  haul  applies. 

But  the  product  of  AX  by  the  number  of  cubic  yards  in  AC 

is  equal  to  the  sum  of  the  products  obtained  by  multiplying  the 

contents  of  each  prismoid  in  AC  by  the  distance  of  its  own 

centre  of  gravity  from  A.    The  distance  of  the  centre  of  gravity 

of  a  prismoid  from  its  mid-section  is  expressed  by  the  formula 

_  I*  (A  -A) 

•    12X27S 

If  we  replace  8  by  its  approximate  value,         J      —  ,  which 


(861> 


will  produce  no  important  error  in  this  case,  we  have 

A  -A' 


CALCULATION    OF    EARTHWORK.  245 

in  which  A  should  always  represent  the  more  remote  end  area 
from  the  starting  point  A,  fig.  106.  Hence,  x  may  be  -j-  or  — , 
and  it  must  be  applied,  with  its  proper  sign,  to  the  distance  of 
the  mid-section  from  the  starting  point  A,  before  multiplying 
by  the  contents  8.  Each  partial  product  is  thus  obtained. 

By  a  similar  process  with  respect  to  the  prismoids  composing 
the  mass  BD,  and  using  the  point  B  as  the  starting  point,  we 
obtain  finally  a  sum  of  the  products  representing  this  portion 
of  the  haul. 

If  a  cut  is  divided,  and  parts  are  carried  in  opposite  direc- 
tions, the  calculation  of  each  part  terminates  at  the  dividing 
line.  If  a  portion  of  the  material  in  AC  is  wasted,  it  must  be 
deducted,  and  the  haul  calculated  only  on  the  remainder. 

The  specified  limit  is  sometimes  made  as  low  as  100  feet, 
sometimes  as  high  as  1000  feet.  A  limit  of  about  300  feet,  how- 
ever is  usually  most  convenient,  as  it  includes  the  wheelbarrow 
work,  and  a  large  part  of  the  carting,  while  it  protects  the  con- 
tractor on  such  long  hauls  as  may  occur. 


27O.  The  Final  Estimate  is  a  complete  statement  in 
detail,  of  the  amount  of  work  done  and  materials  provided,  in 
the  construction  of  the  road,  and  is  the  basis  of  final  settlement 
between  the  company  and  contractor.  Its  preparation  should 
be  begun  as  soon  as  possible  after  the  work  is  in  progress,  and 
should  be  continued,  as  fast  as  the  necessary  data  are  accumu- 
lated, while  the  circumstances  are  still  fresh  in  mind,  and  when 
any  omissions  in  the  field  notes  may  be  readily  supplied.  The 
content  of  each  prismoid,  the  classification  of  its  material,  and 
the  length  of  haul  to  which  it  is  subject,  should  be  matters  of 
special  record  in  a  book  provided  for  that  purpose.  These  re- 
sults having  been  carefully  computed  by  exact  methods  form 
a  standard  of  comparison  for  those  approximate  results  which 
must  be  had  from  time  to  time  during  the  progress  of  the  work, 
and  furnish  a  limit  to  the  amounts  of  the  monthly  estimates. 
The  same  remark  applies  to  all  other  items  of  labor  and  mate- 
rial. The  notes  and  record  of  the  final  estimate  should  be  par- 
ticularly full  and  exact  in  respect  to  all  such  items  as  will  be 
inaccessible  to  measurement  at  the  completion  of  the  work, 
such  as  foundation  pits,  foundation  courses  of  masonry,  cul- 
verts, and  works  under  water. 


246  FIELD 

271.  Monthly  Estimates.— On  or  before  the  last  day 
of  every  month  during  the  progress  of  construction,  measure- 
ments are  taken  to  determine  the  total  amount  of  work  done 
and  material  provided  up  to  that  date.  The  estimates  based 
on  these  measurements  are  called  Monthly  Estimates.  It  is  fre- 
quently necessary  to  take  measurements  for  both  monthly  and 
final  estimates  at  other  times  than  the  end  of  the  month,  as  in 
the  case  of  foundations  which  are  not  long  accessible.  With 
respect  to  each  piece  of  work  satisfactorily  completed,  the 
monthly  estimate  should  be  exact,  and  identical  in  amount 
with  the  final  estimate.  With  respect,  however,  to  items  of 
work  in  progress  at  the  time  of  measurement,  the  monthly 
estimate  is  only  approximate,  yet  should  be  as  precise  as  the 
nature  of  the  case  will  allow;  and  the  quantities  stated  should 
not  be  in  excess  of  fair  proportion  of  the  total  quantities  given 
on  the  final  estimate  for  the  same  piece  of  work. 

A  special  field  book  is  devoted  to  monthly  estimate 
notes.  Each  page  should  be  dated  with  the  day  on  which  the 
notes  upon  it  were  taken.  The  notes  consist  of  measurements 
of  all  sorts,  principally  of  cross  sections  partially  excavated. 
These  sections  should  be  at  the  same  points  on  the  line  as  the 
original  sections,  so  that  comparisons  may  be  made.  Where- 
ever  the  excavation  is  finished  to  grade,  it  is  only  necessary  to 
write  "  completed "  opposite  such  stations,  and  the  quantities 
may  be  taken  from  the  final  estimate  or  computed  from  the 
original  notes.  It  is  frequently  necessary  to  retrace  portions 
of  the  centre  line  in  taking  estimate  notes,  so  that  all  the  field 
instruments  are  required,  but  a  party  of  three  or  four  men  is 
usually  sufficient. 

If  the  contractor  has  provided  materials,  such  as  stone,  lum- 
ber, etc.,  which  are  not  as  yet  put  into  any  structure  when  the 
estimate  is  taken,  these  should  be  measured  and  entered  under 
the  head  of  temporary  allowance,  an  arbitrary  price  be- 
ing used  somewhat  below  the  actual  value  of  the  material  as 
delivered.  Such  allowances  should  never  be  copied  from  one 
month's  estimate  to  the  next,  but  made  anew  on  such  material 
as  may  be  found  that  seems  to  require  it.  But  all  completed 
items  of  contract  work,  and  of  extra  work  when  ordered  by 
the  engineer,  are  necessarily  copied  from  one  monthly  esti- 
mate to  the  next  during  the  continuance  of  the  contract. 

A  blank  form  is  used  by  the  resident  engineer  in  report 


TOPOGRAPHICAL  SKETCHING.  247 

ing  monthly  estimates,  on  which  a  column  is  provided  for  each 
class  of  material  and  work  required  by  the  contract,  while  the 
several  lines,  headed  by  the  numbers  of  the  proper  stations,  are 
devoted  to  the  different  cuttings,  structures,  etc.,  in  consecu- 
tive order  as  they  occur  on  the  line  of  road.  The  estimates  are 
made  out  and  reported  separately  for  the  several  sections  into 
which  the  line  of  road  is  divided  for  letting. 

These  reports  are  reviewed  by  the  division  engineer,  and 
the  footings  copied  upon  another  blank,  which  is  the  monthly 
estimate  proper;  the  prices  are  attached  to  the  items,  and  the 
amounts  extended  and  summed  up.  This  sum  indicates  ap- 
proximately the  total  amount  earned  by  the  contractor  up  to 
date,  from  which  is  deducted  a  certain  percentage  (usually  15 
per  cent.),  which  is  retained  by  the  company  until  the  comple- 
tion of  the  contract.  From  the  remainder  is  deducted  the 
amount  of  previous  payments,  which  leaves  the  amount  due 
the  contractor  on  the  present  estimate.  A  blank  form  of  re- 
ceipt is  appended,  to  be  signed  by  the  contractor.  » 


CHAPTER  XI. 
TOPOGRAPHICAL  SKETCHING. 

272.  Topographical  sketches  taken  on  preliminary  surveys 
are  usually  of  great  value  in  projecting  a  line  for  location; 
they  should  be  made  therefore  as  accurate  and  complete  as 
possible.  In  too  many  instances  sketches  are  presented  having 
a  picturesque  appearance,  but  conveying  little  information,  if 
not  tending  to  mislead  the  map-maker.  The  aim  of  the  topog- 
rapher should  be  to  record  the  topographical  features  either 
side  of  the  line  with  as  much  precision  as  those  directly  upon 
the  line,  without  taking  actual  measurements,  except  in  rare 
instances.  The  eye  and  the  judgment  must  be  usually  depended 
on  for  distances  and  dimensions.  The  sketch  of  a  tract  ex- 
tending to  400  feet  each  side  of  the  line  ought  to  be  accurate 
enough  to  warrant  its  being  copied  literally  upon  the  map.  If 
a  much  wider  range  is  required  it  may  be  advisable  to  use  the 
plane-table;  but  an  approximation  to  plane-table  methods  may 
be  employed  in  ordinary  sketching. 


248  FIELD   ENGINEERING. 

273.  As  artificial  features  are  the  most  readily  de- 
fined and  located  these  should  first  receive  attention  in  making 
a  sketch.     When  recorded  they  form  a  skeleton  upon  which 
the  natural  features  can  be  drawn  with  more  precision  than  if 
the  order  were  reversed.     The  point  where  each  fence  crosses 
the  line  and  the  angle  between  the  two  may  be  sketched  exact- 
ly.    The  distance  along  the  fence  to  any  object  may  be  esti- 
mated, and  checked  (in  case  of  an  oblique  angle)  by  observing 
where  a  line  from  the  object  perpendicular  to  the  centre  line 
would  intersect  the  latter.     The  book  may  be  rested  on  any 
support,  the  centre-line  of  the  page  coinciding  with  the  line  of 
survey,  and  the  direction  of  objects  defined  by  a  small  ruler 
laid  on  the  page.    This  operation  being  repeated  from  another 
point  gives  intersections  which  locate  the  several  objects  on 
the  sketch.     If  the  bearings  are  taken  they  may  be  plotted  on 
the  page  as  well  as  recorded,  giving  the  same  results.     The 
eye  may  be  trained  to  estimate  distances  correctly  by  observ- 
ing the  appearance  of  objects  along  the  measured  line,  the  dis- 
tances to  which  are  therefore  known. 

274.  After  the  artificial  objectslhe  more  distinct  natural 
features  are  to  be  sketched,  as  streams,  shores,  margins  of 
swamps,  forests,  etc.,  ravines,  ridges,  and  bluffs,  taking  care 
that  all  these  outlines  intersect  the  features  of  the  sketch 
already  delineated  at  the  proper  points.     The  correct  repre- 
sentation of  contours  is  the  most  difficult  part  of  sketching, 
since  these  lines  are  quite  imaginary,  yet  for  railroad  maps 
they  are  usually  as  important  as  any  others.     It  is  desirable  to 
know  not  only  the  locality  of  a  hill  or  slope,  but  also  its  shape, 
steepness,  and  height.     This  information  is  best  given  by  con- 
tour lines.    A  contour  is  the  intersection  of  the  surface  of  the 
ground  by  an  imaginary  level  surface.     When  the  surface  is 
real,  like  that  of  a  lake,  the  intersection  is  called  a  shore.     If 
the  water  should  rise  a  certain  height  a  new  shore  would  be 
defined,  and   rising  double  that  height  still  another    shore 
would  result,  each  of  which,  on  the  subsidence  of  the  water, 
would  be  a  contour.    A  practiced  eye  is  able  to  follow  on  the 
ground  the  course  of  a  contour  with  all  its  windings;  but  in 
sketching  them  due  allowance  must  be  made  for  the  fore- 
shortening effect  of  distance.     All  contours  on  the  same  sketch 
should  have  the  same  vertical  interval,  so  that  by  counting 


TOPOGRAPHICAL   SKETCHING.  249 

them  the  height  of  the  hill  may  be  known.  The  spaces  on  the 
sketch  between  contours  vary  as  the  cotangent  of  the  slope 
angle,  so  that  the  width  of  the  spaces  indicates  the  degree  of 
steepness.  The  contours  nearest  the  topographer  should  gene- 
rally be  sketched  first,  although  if  there  be  a  shore  that  is  apt  to 
be  the  best  guide  to  the  shape  of  the  slopes.  If  the  height  of  the 
hill  is  known  and  the  upper  contour  located,  the  other  contours 
can  be  spaced  between  with  less  difficulty,  the  proper  number 
being  ascertained  by  dividing  the  height  by  the  assumed  verti- 
cal interval.  A  special  line  of  levels  up  an  inclined  ravine  or 
sloping  ridge  to  fix  the  contour  points  is  often  of  the  greatest 
service  in  obtaining  correct  results.  Other  random  lines  are 
sometimes  run  to  locate  the  contours  more  definitely.  These 
should  be  made  to  cross  several  contours  rather  than  to  trace  a 
single  one.  Old  preliminary  lines  which  have  proved  useless 
in  themselves  often  furnish  by  their  profiles  valuable  informa- 
tion in  respect  to  contours. 

The  use  of  hatchings  should  be  avoided  in  the  sketch-book, 
except  to  represent  precipitous  banks,  or  slight  terraces,  which 
would  not  be  sufficiently  defined  by  the  contour  system. 
Hatchings  freely  used  consume  too  much  time,  and  fail  to  give 
an  accurate  idea  of  either  slope  or  height,  while  they  obscure 
the  page  for  the  representation  of  other  objects. 

275.  The  centre  line  on  the  page  is  straight,  and  for 
sketching  purposes  the  surveyed »line  on  the  ground  is  assumed 
to  be  so  also.  Slight  deflections  in  the  course  of  a  preliminary 
line  may  be  ignored  in  the  sketch ;  but  if  a  large  angle  occurs 
it  is  better  to  terminate  the  sketch  with  the  course,  and  begin 
again,  leaving  a  few  blank  lines  between  the  two  .sketches. 
On  a  located  line  with  curves,  the  sketch  is  continuous.  The 
curved  line  in  the  field  is  represented  by  the  straight  line  on 
the  page,  and  the  radial  lines  through  the  stations  are  repre- 
sented by  the  parallel  lines  ruled  across  the  page.  All  objects 
are  sketched  at  the  proper  offset  distance  by  scale,  from  the 
centre  line;  but  longitudinally  the  sketch  is  necessarily  dimin- 
ished outside  of  the  curve,  and  magnified  inside  of  the  curve, 
Consequently  topographical  lines  which  are  straight  in  fact  ap- 
pear curved  in  the  sketch,  concave  to  the  centre  line  if  inside 
the  curve,  and  convex  if  outside  of  it.  Such  features  are  cor- 
rectly sketched  by  means  of  offsets  estimated  or  measured 


250  FIELD 


from  each  station  of  the  curve  on  the  radial  lines.  This  kind 
of  distortion  creates  no  confusion  if  properly  done,  for  in  mak- 
ing the  map,  after  drawing  the  curve  and  the  radial  lines,  the 
same  offsets  will  give  the  correct  positions  of  the  objects  delin- 
eated. This  method  is  preferable  to  drawing  a  curved  line  on 
the  page  to  represent  the  centre  line,  as  it  is  difficult  to  draw 
it  correctly;  it  will  cross  the  ruled  lines  obliquely,  rendering 
them  of  no  service  for  offsets  or  scale,  and  moreover  is  likely 
to  run  off  the  page  altogether. 


CHAPTER  XII. 
ADJUSTMENT  OF  INSTRUMENTS. 

Every  adjustment  consists  of  two  processes:  first  the  test,  and 
second  the  correction.  Inasmuch  as  the  amount  of  correction 
is  made  by  estimation,  the  test  must  always  be  repeated  until 
no  further  lack  of  adjustment  is  observable. 

276.  THE  TRANSIT. 

Tlie  level  tubes  should  be  parallel  to  the 
vernier  plate. 

Test :  Place  the  tubes  in  position  over  the  levelling  screws, 
and  turn  the  latter  till  the  hubbies  are  centred;  revolve  the 
plate  180°.  The  bubbles  should  remain  centred;  if  they  have 
retreated — 

Correction  :  Bring  them  half  way  back  to  the  centre  by 
turning  the  adjusting  screws  which  attach  the  tubes  to  the 
plate. 

The  line  of  collimation  should  be  perpendi- 
cular to  the  horizontal  axis. 

Test:  Clamp  the  limb,  and  by  the  tangent  screws  bring 
the  intersection  of  the  cross-hairs  to  cover  a  well-defined  point 
about  on  a  level  with  the  telescope ;  plunge  the  telescope  to 
look  in  the  opposite  direction,  and  note  any  point  about  on  a 
level  with  the  telescope  and  about  equidistant  with  the  first 
point,  which  the  intersection  of  the  cross-hairs  now  happens  to 
cover.  Now  unclamp  the  limb  and  turn  through  180°,  and 
repeat  the  above  operation,  using  the  same  first  point  as  before. 


ADJUSTMENT  OF   INSTRUMENTS.  251 

The  third  point  obtained  should  be  identical'  with  the 
second;  if  not — 

Correction  :  Move  the  vertical  cross-hair  over  one  fourth 
of  the  apparent  distance  from  the  third  to  the  second  point,  by 
turning  the  adjusting  screws  at  the  side  of  the  telescope. 

The  horizontal  axis  should  be  parallel  to  the 
vernier  plate. 

Test :  After  completing  the  above  adjustments  level  the 
limb,  clamp  it,  and  bring  the  intersection  of  the  cross-hairs  to 
cover  some  high  point  so  that  the  telescope  may  be  elevated  to 
a  large  angle;  depress  the  telescope  and  note  some  point  on  the 
ground  now  covered  by  the  intersection  of  the  cross-hairs. 
Now  unclamp  the  limb,  turn  it  through  180°,  and  repeat  the 
above  operation,  using  the  same  high  point  as  before.  The 
third  point  found  should  be  identical  with  the  second;  if  not — 

Correction  :  Kaise  the  end  of  the  axis  opposite  the  second 
point  (or  lower  the  other  end)  by  a  small  amount,  by  turning 
the  adjusting  screws  in  the  standard.  The  amount  of  motion 
required  is  only  determined  by  repeated  trials  until  the  test  is 
satisfied. 

The  intersection  of  the  cross-hairs  should 
appear  in  the  centre  of  the  field  of  view. 

Test :  Bring  the  cross-hairs  into  focus  and  direct  the  tele- 
scope toward  the  sky,  or  hold  a  sheet  of  blank  paper  in  front 
of  it.  If  the  intersection  appear  eccentric — 

Correction  :  Turn  the  screws  (by  pairs)  which  support 
the  end  of  the  eyepiece  until  the  desired  result  is  obtained. 

If  there  be  a  level  on  the  telescope  it  should  be 
parallel  to  the  line  of  collimatioii. 

Drive  two  stakes  equidistant  from  the  instrument  in  exactly 
opposite  directions,  and  having  perfected  the  previous  adjust- 
ments, level  the  plate  carefully,  clamp  the  telescope  in  about 
a  horizontal  position,  and  observe  a  rod  placed  on  each  stake. 
Have  the  stakes  driven  by  trial  until  the  rod  reads  alike  on 
both.  The  heads  of  the  stakes  are  then  on  a  level.  Re- 
move the  instrument  beyond  one  stake,  and  set  it  up  in 
line  with  the  two,  level  the  plate,  and  elevate  or  depress  the 
telescope  to  a  position  which  will  again  give  equal  readings 
on  the  stakes.  The  line  of  collimation  is  now  level — 


252  FIELD    ENGINEERING. 

Test :  "While  in  this  position  the  bubble  of  the  attached 
level  should  stand  centred;  if  not — 

Correction :  Bring  the  bubble  to  the  centre  by  turning 
the  nuts  at  one  end  of  the  tube,  while  the  cross-hair  continues 
to  give  equal  readings. 

277.  THE  Y  LEVEL. 

The  line  of  collimation  should  coincide  with 
the  axis  of  the  telescope. 

Test :  Clamp  the  spindle,  and  bring  the  intersection  of  the 
cross-hairs  to  cover  a  well-defined  point  by  the  tangent  and 
levelling  screws;  revolve  the  telescope  half  over  in  the  Ys,  so 
that  the  level  tube  is  on  top.  The  intersection  of  the  cross- 
hairs should  still  cover  the  point.  If  either  hair  has  departed — 

Correction  :  bring  it  half  way  back  by  means  of  the  pair 
of  adjusting  screws  at  the  extremities  of  the  other  hair. 

The  attached  .level  should  be  parallel  to  the 
axis  of  the  telescope. 

Test :  Bring  the  telescope  over  one  pair  of  levelling  screws, 
clamp  the  spindle,  open  the  clips,  and  bring  the  bubble  to  the 
centre.  Then  gently  remove  the  telescope  from  the  Ys,  and 
replace  it  end  for  end.  If  the  Ys  have  not  been  disturbed,  the 
bubble  should  return  to  the  centre.  If  it  does  not — 

Correction  :  bring  the  bubble  half  way  back  by  turning 
the  nuts  at  one  end  of  the  tube. 

But  as  now  the  level  tube  and  telescope  may  only  lie  in 
parallel  planes,  and  yet  not  be  parallel  to  each  other— 

Test:  bring  the  bubble  to  the  centre  as  before,  and  turn 
the  telescope  on  its  axis  so  as  to  bring  the  level  tube  out  to  one 
side.  The  bubble  should  remain  centred.  If  it  has  departed — 

Correction  :  bring  it  back  to  the  centre  by  the  adjusting 
screws  at  one  end. 

The  axis  of  the  telescope  should  be  at  rig-ht 
angles  to  the  spindle. 

Test :  Having  completed  the  above  adjustments  (and  not 
before),  fasten  down  the  clips,  unclamp  the  spindle,  and  bring 
the  bubble  to  the  centre  over  each  pair  of  levelling  screws  in 
succession,  then  swing  the  telescope  end  for  end  on  the  spin- 
dle. The  bubble  should  settle  at  the  centre.  If  it  do  not- 
Correction  :  bring  it  half  way  back  by  the  large  nuts  at 
one  end  of  the  bar. 


EXPLANATION   OF   TABLES.  253 

278.  THE  THEODOLITE, 

This  instrument  being  a  combination  of  Transit  and  Level, 
its  several  adjustments  are  to  be  made  according  to  the  rules 
already  given  for  those  instruments. 


CHAPTER  XIII. 
EXPLANATION  OF  TABLES. 

TABLE  I. — Contains  concise  statements  of  such  geometrical 
truths  as  are  applicable  to  the  various  discussions  in  this  volume. 
References  are  given  to  Dayies'  Geometry,  in  which  the  demon- 
strations of  the  propositions  majr  be  found. 

TABLE  II. — Contains  all  the  formulae  necessary  to  the  solu- 
tion of  any  plane  triangle ;  also,  a  select  list  of  miscellaneous 
formulas.  A  few  formulae  with  respect  to  the  versed  sine  and 
external  secant  are  new. 

TABLE  III. — Contains  a  complete  list  of  formulae  expressing 
the  relations  between  the  radius,  tangent,  chord,  versed  sine, 
external  secant,  and  central  angle  of  a  railway  curve;  also,  the 
relations  between  the  radius,  degree  of  curve,  length  of  curve, 
and  central  angle.  The  notation  is  identical  with  that  used 
elsewhere  in  the  book. 

TABLE  IV. — Contains  the  radius,  and  its  logarithm,  for  every 
degree  of  curve  to  single  minutes  up  to  10  degrees,  and  thence 
by  larger  intervals  up  to  50  degrees.  With  the  radius  is  given 
also  the  perpendicular  off-set,  t,  from  the  tangent  to  a  point  on 
the  curve  at  the  end  of  the  first  100-foot  chord  from  the  tan- 
gent-point, and  the  middle  ordinate,  m,  of  a  100-foot  chord. 
See  eqs.  (16,  34,  37,  40,  and  305). 

TABLE  V. — Contains  the  corrections  to  be  added  to  the  tan- 
gents and  externals  of  any  railroad  curve,  as  obtained  by  refe- 
rence to  Table  VI.,  according  to  the  degree  of  the  given  curve 
(found  at  head  of  columns),  and  its  central  angle,  (found  in  the 


254:  FIELD    ENGINEERING. 

first  column.)  If  the  given  degree  of  curve,  or  central  angle, 
does  not  appear  in  the  table,  the  exact  value  of  the  correction 
may  be  easily  obtained  by  interpolation. 

TABLE  VI. — Contains  the  exact  values  of  the  tangents,  T, 
and  externals,  E,  to  a  1  degree  curve,  for  every  10  minutes  of 
central  angle,  from  1°  to  120°  50'.  Approximate  values  of  the 
tangent  and  external  to  any  other  degree  of  curve  may  be  had 
by  simply  dividing  the  tabular  values  opposite  the  given  cen- 
tral angle  by  the  given  degree  of  curve,  expressed  in  degrees. 
These  approximations  may  be  made  exact  by  adding  the  proper 
corrections  taken  from  Table  V.  See  eqs.  (21)  and  (24). 

TABLE  VII. — Contains  the  value  of  Long  Chords  of  from  2 
to  12  stations,  for  every  10  mmutes  of  degree  of  curve  from  0° 
to  15°,  and  of  a  less  number  of  stations  for  degrees  of  curve  be- 
tween 15°  and  30°.  As  the  chord  of  one  station  is  always  100 
feet,  the  column  of  the  first  station  gives  instead  the  length  of 
arc  subtended  by  the  chord  of  100  feet.  See  §§  121, 122,  123, 
124,  125. 

TABLE  VIII. — Contains  the  values  of;  Middle  Ordinates  to 
long  chords  of  from  2  to  12  stations,  for  every  10  minutes  of 
degree  of  curve  from  0°  to  10°,  and  of  from  2  to  6  stations  for 
every  curve  from  10°  to  20°,  at  10-minute  intervals.  The  table 
may  be  used,  not  only  to  fix  the  middle  point  of  an  arc,  but 
also,  in  conjunction  with  the  table  of  long  chords,  to  locate  in- 
termediate stations.  See  §§  121,  122,  123,  124,  125. 

TABLE  IX. — Contains  the  chords  of  a  series  of  angles  vary- 
ing by  half  degrees  up  to  30°  for  radii  varying  by  100  feet  up  to 
1000  feet.  It  shows,  therefore,  the  linear  opening  between 
the  extremities  of  two  equal  lines  at  any  given  number  of  hun- 
dred feet  from  their  intersection,  when  the  angle  does  not  ex- 
ceed 30°.  For  any  distance  exceeding  1000  we  have  only  to 
add  to  the  value  found  in  that  column,  the  value  found  in  the 
column  headed  by  the  excess  of  distance  over  1000  feet.  Con- 
versely, the  table  gives  the  angular  deflection  required  between 
two  equal  lines,  in  order  that  at  a  given  distance  from  the  point 
of  intersection  they  may  be  separated  a  given  amount. 


EXPLANATION    OF   TABLES.  255    - 


TAULE  X. — 1.  Contains  values  of  the  ratio  u  —  — ,  accord - 

A 

ing  to  the  notation  of  §  147  for  finding  the  angle  i  (Fig.  34) 
between  the  radius  PO  of  the  curve  at  any  point  P,  and  the 
tangent  PK  to  the  valvoid  arc  PX  by  the  simple  formula 
eq.  (80)£=?£A.  The  table  embraces  lengths  of  curve  from 
300  to  2000  feet,  and  central  angles  from  10°  to  120°. 

When  —    —  =  60°  u  =  $,  and  for  hasty  approximation  this 
1000 

value  of  u  may  be  assumed  in  any  case  without  consulting 
the  table. 

T 

2.  Contains  values  of  the  ratio  «  =  — -  for  finding  the  radius 

of  the  valvoid  arc  at  the  point  P  (Fig.  35)  in  terms  of  the 
length  of  curve  L  =  AP  by  the  simple  formula,  eq.  (82), 
r  =  vL. 

3.  Contains  values  of  the  length  I,  of  a  valvoid  arc  limited 
by  two  curves  of  equal  length  laid  out  from  the  same  tangent 
and  same  P.  C. ,  but  whose  central  angles  djft'er  by  1°.     The 
length  L  of  each  curve  is  given  in  the  first  column,  and  the 

half  sum  of  their  central  angles  I—  ~ I  is  given  at  the 

head  of  the  other  columns. 

When  the  central  angles  of  two  curves  of  equal  length 
differ  by  x  degrees  the  length  I  of  the  valvoid  arc  Joining 
their  extremities  is  expressed  by  the  simple  formula,  Fig.  36, 

cq.  (86)  l=P'P"  =(A'  -  A"K 

in  which  lt  is  taken  from  the  column  headed  by  -  — ~— 

and  opposite  the  given  value  of  Z;or  lt  is  found  by  inter- 
polation  if  necessary.  See  §  150  and  example. 

TABLE  XI. — Contains  the  measurements  necessary  to  lay 
down  a  turnout  with  frogs  of  given  numbers  or  angles  for 
both  a  standard  and  a  three-foot  gauge.  The  distance  BF  is 
measured  on  the  rail  of  the  given  track  from  the  heel  of  the 
switch  to  the  point  of  the  frog,  while  of  is  the  chord  of  the 
centre  line  of  the  turnout  between  the  same  points.  The 
radius  r  applies  to  the  centre  line  of  the  turnout.  The  dis- 
tance aF"  is  measured  on  the  centre  line  of  the  straight  track 


256  FIELD 


from  the  lied  of  the  switch  to  the  point  of  the  middle  frog. 
The  length  of  switch  AD  should  conform  to  the  tabular 
values  unless  the  throw  is  to  he  different  from  that  assumed 
in  the  table.  See  §§  180,  181,  182. 

TABLE  XII.  —  Contains  the  middle  ordinates  of  chords  vary- 
ing in  length  from  10  to  32  feet,  and  for  degrees  of  curve  vary- 
ing from  1°  to  50°.  The  use  of  the  table  is  obvious.  See  §  199. 

TABLE  XIII.  —  Gives  the  proper  difference  in  elevation  of 
rails  on  curves  of  various  degrees  from  1°  to  50°  for  veloci- 
ties varying  from  10  to  60  miles  per  hour.  See  §  201. 

TABLE  XIV.  —  Gives  the  rise  of  grades  in  feet  per  mile  and 
their  angle  of  inclination  corresponding  to  a  rise  per  station 
(100  feet)  varying  from  0.01  foot  to  10  feet. 

TABLE  XV.  —  Contains  values  of  the  formula  (log  h  —  1) 
60384.3  in  which  li  =  reading  of  the  barometer  in  inches.  The 
inches  and  tenths  of  the  readings  are  in  the  left-hand  column, 
while  the  hundredths  are  found  at  the  top  of  the  other  columns. 
The  difference  of  any  two  values  corresponding  to  two  read- 
ings taken  simultaneously  at  any  two  stations  is  the  differ- 
ence in  elevation  in  feet  of  those  stations.  But  the  differ- 
ence in  height  so  found  is  subject  to  a  correction  for  tempera- 
ture given  in  the  next  table.  See  §  10. 

TABLE  XVI.  —  Contains  coefficients  of  correction  for  atmos- 
pheric temperature,  by  which  the  approximate  heights  ob- 
tained by  Table  XV.  are  to  be  multiplied  for  a  correction  of 
these  heights,  which  correction  is  to  be  added  or  subtracted 
according  as  the  coefficient  given  in  the  table  is  marked 
-f  or  -.  See  §  11. 

TABLE  XVII.  —  Contains  corrections  in  feet,  required  by  the 
curvature  of  the  earth  and  the  refraction  of  the  atmosphere,  to 
be  applied  to  the  elevation  of  a  distant  object  as  obtained  by  a 
level  or  theodolite  observation  for  distances  ranging  from 
300  feet  to  10  miles.  See  §  119. 

TABLE  XVIII.  —  Contains  the  coefficients  for  reducing  the 
space  on  a  vertical  rod  intercepted  by  the  stadia  hairs  when 


EXPLANATION   OF  TABLES.  257 

the  line  of  collimation  is  inclined  to  the  horizon,  to  the  space 
that  would  be  intercepted  were  the  line  of  collimation  horizon- 
tal; provided,  that  the  visual  angle  0  denned  by  the  stadia  hairs 
is  such  that  tan  ^0  =  .005  or  fl  =  0°  34  22". 63,  which  is  its 
customary  value  in  surveying  instruments.  The  angle  of  in 
clination  a  is  taken  at  every  10  minutes  through  half  a  quad- 
rant. 

TABLE  XIX. — Contains  the  logarithms  of  the  coefficients 
given  in  Table  XVIII. 

TABLE  XX. — Gives  the  lengths  of  circular  arcs  to  a  radius 
=  1. 

To  find  the  length  of  any  arc  expressed  in  degrees,  minutes, 
and  seconds,  take  from  the  table  the  lengths  of  the  given  num- 
ber of  degrees,  minutes,  and  seconds  respectively,  and  multi- 
ply their  sum  by  the  length  of  the  radius.  The  product  is  the 
length  of  arc  required. 

TABLE  XXI. — Contains  the  values  of  minutes  and  seconds 
expressed  in  decimals  of  a  degree,  for  every  10  seconds  of  arc, 
and  also  for  quarter  minutes  up  to  one  degree. 

TABLE  XXII. — Contains  the  values  of  inches  and  fractions 
expressed  in  decimals  of  a  foot  for  every  32d  of  an  inch  up  to 
one  foot. 

TABLE  XXIII. — Contains  the  squares,  cubes,  square  roots, 
cube  roots,  and  reciprocals  of  numbers  from  1  to  1054.  Its 
use  may  be  greatly  extended  by  observing  that  if  any  number 
is  multiplied  by  n  its  square  is  multiplied  by  »2,  its  cube  by 

n3,  and  its  reciprocal  by  — .  ,  .  . 

TABLE  XXIV. — The  logarithm  of  a  number  consists  of 
two  parts,  a  whole  number  called  the  characteristic,  and  a  deci- 
mal called  the  mantissa.  All  numbers  which  consist  of  the 
same  figures  standing  in  the  same  order  have  the  same  man- 
tissa, regardless  of  the  position  of  the  decimal  point  in  the 
number,  or  of  the  number  of  ciphers  which  precede  or  follow 
the  significant  figures  of  the  number.  The  value  of  the  char- 
acteristic depends  entirely  on  the  position  of  the  decimal  point 
in  the  number.  It  is  always  one  less  than  the  number  of 


258  FIELD   ENGINEERING. 

figures  in  the  number  to  the  left  of  the  decimal  point.  The 
value  is  therefore  diminished  by  one  every  time  the  decimal 
point  of  the  number  is  removed  one  place  to  the  left,  and  vice 
versa.  Thus 

Number.  Logarithm. 

13840.  4.141136 

1384.0  3.141136 

138.40  2.141136 

13.84  1.141138 

1.384  0.141136 

.1384  - 1.141136 

.01384  -2.141136 

.001384  -3.141136 

etc.  etc. 

The  mantissa  is  always  positive  even  when  the  characteristic 
is  negative.  We  may  avoid  the  use  of  a  negative  characteristic 
by  arbitrarily  adding  10,  which  may  be  neglected  at  the  close 
of  the  calculation.  By  this  rule  we  have 

Number.  Logarithm. 

1.384  0.141136 

.1384  9.141136 

.01384  8.141136 

.001384  7.141136 

etc,  etc. 

No  confusion  need  arise  from  this  method  in  finding  a  number 
from  its  logarithm;  for  although  the  logarithm  6.141136  repre- 
sents either  the  number  1,384,000,  or  the  decimal  .0001384,  yet 
these  are  so  diverse  in  their  values  that  we  can  never  be  uncer- 
tain in  a  given  problem  which  to  adopt. 

The  table  XXIV.  contains  the  mantissas  of  logarithms,  car- 
ried to  six  places  of  decimals,  for  numbers  between  1  and  9999, 
inclusive.  The  first  three  figures  of  a  number  are  given  in  the 
first  column,  the  fourth  at  the  top  of  the  other  columns.  The 
first  two  figures  of  the  mantissa  are  given  only  in  the  second 
column,  but  these  are  understood  to  apply  to  the  remaining 
four  figures  in  either  column  following,  which  are  comprised 
between  the  same  horizontal  lines  with  the  two. 

If  a  number  (after  cutting  off  the  ciphers  at  either  end)  con- 
sists of  not  more  than  four  figures,  the  mantissa  may  be  taken 
direct  from  the  table ;  but  by  interpolation  the  logarithm  of  a 
number  having  six  figures  may  be  obtained.  The  last  column 
contains  the  average  difference  of  consecutive  logarithms  on 


EXPLANATION    OF  TABLES.  259 

the  same  line,  but  for  a  given  case  the  difference  needs  to  be 
verified  by  actual  subtraction,  at  least  so  far  as  the  last  figure 
is  concerned.  The  lower  part  of  the  page  contains  a  complete 
list  of  differences,  with  their  multiples  divided  by  10. 

To  find  the  logarithm  of  a  number  having-  six 
figures  :— Take  out  the  mantissa  for  the  four  superior  places 
directly  from  the  table,  and  find  the  difference  between  this 
mantissa  and  the  next  greater  in  the  table.  Add  to  the  man- 
tissa taken  out  the  quantity  found  in  the  table  of  proportional 
parts,  opposite  the  difference,  and  in  the  column  headed  by  the 
fifth  figure  of  the  number;  also  add  ^  the  quantity  in  the  col- 
umn headed  by  the  sixth  figure.  The  sum  is  the  mantissa 
required,  to  which  must  be  prefixed  a  decimal  point  and  the 
proper  characteristic. 

•Example.— Find  the  log  of  23.4275. 

For  2342  mantissa  is  369587 

"     diff.  185  col.  7  129.5 

"       "       "     "    5  9.2 


Ans.  For  23.4275  log  is    1.369726 

The  decimals  of  the  corrections  are  added  together  to  deter- 
mine the  nearest  value  of  the  sixth  figure  of  the  mantissa. 

To  find  the  number  corresponding  to  a  given 
logarithm. — If  the  given  mantissa  is  not  in  the  table  find  the 
one  next  less,  and  take  out  the  four  figures  corresponding  to  it; 
divide  the  difference  between  the  two  mantissas  t>y  the  tabu- 
lar difference  in  that  part  of  the  table,  and  annex  the  figures  of 
the  quotient  to  the  four  figures  already  taken  out.  Finally, 
place  the  decimal  point  according  to  the  rule  for  characteristics, 
prefixing  or  annexing  ciphers  if  necessary.  The  division  re- 
quired is  facilitated  by  the  table  of  proportional  parts,  which 
furnishes  by  inspection  the  figures  of  the  quotient. 

Example. — Find  the  number  of  which  the  logarithm  is 
8.263927  8.2(53927 

First  4  figures  1836  from  263873 

Diff.          5To 
Tabular  diff.  =236          .-.  5th  fig.  =2  47.2 


6.80 
6th  fig.  =  3  7.08 

Ans.  No.  =  .0188623  or  183,623,000. 


2GO  FIELD   ENGINEERING. 

The  number  derived  from  a  six-place  logarithm  is  not 
reliable  beyond  the  sixth  figure. 

At  the  end  of  table  XXIV.  is  a  small  table  of  logarithms  of 
numbers  from  1  to  100,  with  the  characteristic  prefixed,  for 
easy  reference  when  the  given  number  does  not  exceed  two 
digits.  But  the  same  mantissas  may  be  found  in  the  larger 
table. 

TABLE  XXV. — The  logarithmic  sine,  tangent, 
etc.  of  an  arc  is  the  logarithm  of  the  natural  sine,  tangent, 
etc.  of  the  same  arc,  but  with  10  added  to  the  characteristic  to 
avoid  negatives.  This  table  gives  log  sines,  tangents,  cosines, 
and  cotangents  for  every  minute  of  the  quadrant.  With  the 
number  of  degrees  at  the  left  side  of  the  page  are  to  be  read 
the  minutes  in  the  left-hand  column  ;  with  the  degrees  on 
the  right-hand  side  are  to  be  read  the  minutes  in  the  right-hand 
column.  When  the  degrees  appear  at  the  top  Qf  the  page  the 
top  headings  must  be  observed,  when  at  the  bottom  those  at 
the  bottom.  Since  the  values  found  for  arcs  in  the  first  quad- 
rant arc  duplicated  in  the  second,  the  degrees  are  given  from 
0°  to  180°.  The  differences  in  the  logarithms  due  to  a  change 
of  one  second  in  the  arc  are  given  in  adjoining  columns. 

To  find  the  log.  sin,  cos,  tan,  or  cot  of  a  given 
arc. :  Take  out  from  the  proper -column  of  the  table  the  log- 
arithm corresponding  to  the  given  number  of  degrees  and 
minutes.  If  there  be  any  seconds  multiply  them  by  the  ad- 
joining tabular  difference,  and  apply  their  product  as  a  cor- 
rection to  the  logarithm  already  taken  out.  The  correction  is 
to  be  added  if  the  logarithms  of  the  table  are  increasing  with 
the  angle,  or  subtracted  if  they  are  decreasing  as  the  angle  in- 
creases. In  the  first  quadrant  the  log  sines  and  tangents  in- 
crease, and  the  log.  cosines  and  cotangents  decrease  as  the 
angle  increases. 

Example.— Find  the  log  sin  of  9°  28'  20". 

Log  sin  of  9°  28' is  9.216097 

Add  correction  20  X  12.62  252 

Ans.  9.216349 
Exampk..—  Find  the  log  cot  of  9°  28'  20". 

Log  cotan  of  9°  28'  is  10.777948 

Subtract  correction  20  X  12.97  259 


EXPLANATION"   OP  TABLES.  261 

To  find  the  angle  or  arc  corresponding  to  a 
given  logarithmic  sine,  tangent,  cosine,  or  co- 
tangent.— If  the  given  logarithm  is  found  in  the  proper 
column  take  out  the  degrees  and  minutes  directly;  if  not,  find 
the  two  consecutive  logarithms  between  which  the  given 
logarithm  would  fall,  and  adopt  that  one  which  corresponds  to 
the  least  number  of  minutes;  which  minutes  take  out  with  the 
degrees,  and  divide  the  difference  between  this  logarithm  and 
the  given  one  by.  the  adjoining  tabular  difference  for  a  quo- 
tient, which  will  be  the  required  number  of  seconds. 

With  logarithms  to  six  places  of  decimals  the  quotient  is 
not  reliable  beyond  the  tenth  of  a  second. 

Example. — 9.383731  is  the  log  tan  of  what  angle? 
Next  less  9.383682  gives  13°  36' 

Diff.  ~~49.00  -^  9.20  =  05".3 


Ans.     13°  36'  05". 3 

Example. — 9.249348  is  the  log  cos  of  what  angle? 
Next  greater  583  gives  79°  45' 

Diff.  235  -*- 11.65  =  20".2 


Ans.     79°  45  20".  2 

The  above  rules  do  not  apply  to  the  first  two  pages  of  this 
table  (except  for  the  column  headed  cosine  at  top)  because 
here  the  differences  vary  so  rapidly  that  interpolation  made  by 
them  in  the  usual  way  will  not  give  exact  results. 

On  the  first  two  pages,  the  first  column  contains  the  number 
of  seconds  for  every  minute  from  I'to2°;  the  minutes  are 
given  in  the  second,  the  log.  sin.  in  the  third,  and  in  the  fourth 
arc  the  last  three  figures  of  a  logarithm  which  is  the  difference 
between  the  log  sin  and  the  logarithm  of  the  number  of  sec- 
onds in  the  first  column.  The  first  three  figures  and  the  char- 
acteristic of  this  logarithm  are  placed,  once  for  all,  at  the  head 
of  the  column. 

To  find  the  log  sin  of  an  arc  less  than  2°  given 

to  seconds.— Reduce  the  given  arc  to  seconds,  and  take  the 
logarithm  of  the  number  of  seconds  from  the  table  of  loga- 
rithms, and  add  to  this  the  logarithm  from  the  fourth  column 
opposite  the  same  number  of  seconds.  The  sum  is  the  log  sin 
required. 
The  logarithm  in  the  fourth  column  may  need  a  slight  inter- 


262  FIELD    ENGINEERING. 

polation  of  the  last  figure,  to  make  it  correspond  closely  to  the 
given  number  of  seconds. 

Example.— Find  the  log  sm  of  1°  39'  14".  4. 

1°  39'  14".4  =  5954".4  log  3.774838 

add  (q  -  I)  4.685515 

Ans.  log  sin  8.4G0353 

Log  tangents  of  small  arcs  are  found  in  the  same  way,  only 
taking  the  last  four  figures  of  (q  —  1)  from  thejift7i  column. 


Example.— Find  the  log  tan  of  0°  52'  35". 

add  (q  -  I)  4.685609 


52'  35"  =  (3120"  -f  35")  =  3155"  log  3.498999 

—  I) 


Ans.     log  tan  8.184608 

To  find  the  log-  cotangent  of  an  angle  less  than 
2°  given  to  seconds. — Take  from  the  column  headed  ( q-\- 1) 
the  logarithm  corresponding  to  the  given  angle,  interpolating 
for  the  last  figure  if  necessary,  and  from  this  subtract  the  loga- 
rithm of  the  number  of  seconds  in  the  given  angle. 

Example.—  Find  the  log  cotan  of  1°  44'  22". 5. 

q  +  I  15.314292 
6240"  +  22",5  =  6262,5  log    3.796748 

Ans.     11.517544 

These  two  pages  may  be  used  in  the  same  way  when  the 
given  angle  lies  between  88°  and  92  ,  or  between  178°  and  180° ; 
but  if  the  number  of  degrees  be  found  at  the  bottom  of  the  page, 
the  title  of  each  column  will  be  found  there  also;  and  if  the 
number  of  degrees  be  found  on  the  right  hand  side  of  the  page, 
the  number  of  minutes  must  be  found  in  the  right  hand  col- 
umn, and  since  here  the  minutes  increase  upward,  the  number 
of  seconds  on  the  same  line  in  the  first  column  must  be  dimin- 
ished by  the  odd  seconds  in  the  given  angle  to  obtain  the  num- 
ber whose  logarithm  Is  to  be  used  with  (q  ±  I)  taken  from  the 
table. 

Example.— Find  the  log  cos  of  88°  41'  12". 5 

(q  -I)  4,685537 
4740"  -  12".5  =  4727.5  log  3,674631 

Ans.  8.360168 


EXPLANATION   OE   TABLES.  263 

Example.— Find  the  log  tan  of  90°  30'  50". 

q  4- 1  15.314413' 

1800"  +  50'  =  1850'  log  -3.267172 

Ans.    127047241 

To  find  the  arc  corresponding  to  a  given  log 
sin,  cos,  tan,  or  cotan  which  falls  within  the 
limits  of  the  first  two  pages  of  Table  XXV.— 

Find  in  the  proper  column  two  consecutive  logarithms  be- 
tween which  the  given  logarithm  falls.  If  the  title  of  the 
given  function  is  found  at  the  top  of  that  column  read  the 
degrees  from  the  top  of  the  page;  if  at  the  bottom  read  from 
the  bottom.  :M-  . 

Find  the  value  of  (q  —  I)  or  (q  -fr  I),  as  the  case  may  require, 
corresponding  to  the  given  log  (interpolating  for  the  last  figure 
if  necessary).  Then  if  q  —  given  log  and  I  =  log  of  number  of 
seconds,  n,  in  the  required  arc,  we  have  at  once  I  =  q  —  (q  —  I) 
or  I  =  (q  -j-  0  ~~  Q>  whence  n  is  easily  found. 

Find  in  the  first  column  two  consecutive  quantities  between 
which  the  number  n  falls,  and  if  the  degrees  are  read  from 
the  left  hand  side  of  the  page,  adopt  the  less,  take  out  the 
minutes  from  the  second  column,  and  take  for  the  seconds 
the  difference  between  the  quantity  adopted  and  the  number 
n.  But  if  the  degrees  are  read  from  the  right  hand  side  of  the 
page,  adopt  the  greater  quantity,  take  out  the  minutes  on  the 
same  line  from  the  right-hand  column,  and  for  the  seconds 
take  the  difference  between  the  number  adopted  and  the  num- 
ber n. 

Example. — 11.734268  is  the  log  cot  of  what  arc? 

q  +  I  15.314376 

q  11.734268 

.'.     n=  3802.8                                     ~~&580l08 

For  1°  adopt  3780.        giving  03' 

Difference  22".  8 

An*.  1°  03'  22".8  or  178°  56'  37".2. 

Example.—  8.201795  is  the  log  cos  of  what  arc? 

9  -  ^  4.685556 

9  ^8.201795 

.'•       n=  3282".  8                                    3.516239 

For  89°  adopt  3300.       giving  05' 

Difference  17".  2 

Am.  89°  05'  17".2  or  90°  54'  42".8. 


264  FIELD   EffGLffEEBLNTG. 

TABLE  XXYI. — Contains  logarithmic  versed  sines  and  ex- 
ternal secants  for  every  minute  of  the  quadrant,  with  the 
differences  of  the  same  corresponding  to  a  change  of  1  second 
in  the  arc  or  angle.  Interpolation  for  seconds  is  made  in  the 
same  manner  as  with  log  sines  of  the  preceding  table,  except 
on  the  first  two  pages.  For  angles  less  than  4°  the  differences 
vary  so  rapidly  that  interpolation  by  direct  proportion  will  not 
give  exact  values.  . 

On  the  first  two  pages  the  column  headed  q  —  21  contains 
the  difference  between  the  log  versed  sine  (or  log  ex  secant)  of 
an  arc  and  twice  the  logarithm  of  the  number  of  seconds  in  the 
same  arc.  The  characteristic,  and  first  three  decimals  (9.070) 
are  common  to  all  the  logarithms  in  these  columns  up  to  3°  19' 
for  log  vers  sines,  where  it  changes  to  (9.069),  as  shown  at  the 
foot  of  the  column;  and  up  to  4°  for  log  ex  secants,  where  it 
changes  to  (9.071).  At  the  point  of  change  a  cipher  is  replaced 
by  the  mark  +  to  call  attention. 

To  find  the  log  vers  sin,  or  log  ex  sec  of  an 
angle  given  to  seconds. — Reduce  the  angle  to  seconds, 
take  the  logarithm  of  this  number,  multiply  it  by  2,  and  add 
the  product  to  the  logarithm  in  the  column  (q  —  21)  found  op- 
posite the  given  angle.  The  log  (q  —  21)  should  be  corrected 
by  interpolation  for  the  fractional  part  of  a  minute  in  the  given 
angle. 

Example.—  What  is  the  log  ex  secant  of  2°  14'  43". 7? 

2°  14'  43". 7  -  8040"  -f  43.7  =  8083". 7      log  3.907610 

2 


21      7.815220 
(q  -  20    9.070064 

Ans.    .'.  q     6.885284 


To  find  the  arc  corresponding1  to  a  given  log" 
vers,  or  log  ex  sec. — Find  in  the  column  of  log  vers,  or 
log  ex  sec  the  two  values  between  which  the  given  log  falls,, 
and  take  out  from  the  column  (q  —  21)  the  logarithm  corres- 
ponding to  the  given  log  (interpolating  for  the  value  of  the  last 
figure  if  necessary).  Subtract  this  from  the  given  logarithm 
and  divide  by  2.  The  quotient  is  the  logarithm  of  the  num- 
ber of  seconds  in  the  required  arc. 


EXPLANATION   OF   TABLES.  265 

Example. — 7.344728  is  the  log  vers  of  what  arc? 

q  7.344728  • 

3°  48'  -f-  (q  -  21)          9.069960 

2)8.274768 

13720".  9  .-.     I       4.137384 

13680. 


Ans.      3°  48'  40".9 

To  find  the  log  ex  sec  of  an  arc  greater  than 
88°  given  to  seconds. — Take  from  the  column  (q-\-l) 
the  logarithm  corresponding  to  the  given  arc,  interpolating  for 
the  fraction  of  a  minute.  From  this  subtract  the  logarithm  of 
the  number  of  seconds  in  the  complement  of  the  given  arc. 

Example.— What  is  the  log  ex  sec  of  88°  24'  20".  5? 

For  88°  24'  q  -4- 1  15.302183 

on  fi 
Correction  129  X  ~-=  44 


q  +  l        15.302227 
Comp.  88°  24'  20".5  _=  5739".5  log    3.758874 

Ans.     log  ex  sec  11.543353 

To  find  the  angle   corresponding  to  a  given 
log  ex  sec  when  the  angle  is  greater  than  88°.— 

Find  in  the  table  the  two  consecutive  log  ex  secants  between 
which  the  given  one  falls,  and  then  find  by  interpolation  the 
value  of  the  log  (q-\-l)  corresponding  to  the  given  log  ex  sec 
and  subtract  the  latter  from  it.  The  difference  will  be  the 
logarithm  of  the  number  of  seconds  in  the  complement  of  the 
required  angle,  which  is  then  easily  found. 

Example.—  11.924368  is  the  log  ex  sec  of  what  arc? 

Given  log  ex  sec  11.924368 

Next  less  11.918290  ?  +  *        15.309225 


Diff.  6078 


q  +  I        15.309296 
Given  log  ex  sec  Jl.  924368 

0°  40'  26".2  =  2426".2  .'.     log    3.384928 

An*.  89°  19  33".8. 


266  FIELD    EKGIKEEBIKG. 

TABLE  XXVII. — Contains  natural  sines  and  cosines,  to  five 
places  of  decimals  for  every  minute  of  the  quadrant.  Correc- 
tions for  fractions  of  a  minute  are  made  directly  proportional 
to  the*  difference  of  consecutive  values  in  the  table;  positive 
for  sines,  negative  for  cosines. 

TABLE  XXVIII. — Contains  natural  tangents  and  cotangents 
to  five  places  of  decimals  for  every  minute  of  the  quadrant. 
Corrections  for  fractions  of  a  minute  are  made  directly  propor- 
tional to  the  difference  of  consecutive  values  in  the  table  ; 
positive  for  tangents,  negative  for  cotangents. 

TABLE  XXIX. — Contains  natural  versed  sines  and  external 
secants  to  five  places  of  decimals  for  every  minute  of  the 
quadrant.  Corrections  for  fractions  of  a  minute  are  made 
directly  proportional  to  the  difference  of  consecutive  values. 
They  are  positive  in  every  case. 

TABLE  XXX.— Contains  the  number  of  cubic  yards  con- 
tained in  prismoids  of  various  side  slopes,  bases,  and  depths, 
as  indicated  by  the  title  and  the  numbers  in  the  first  column. 
Each  prismoid  is  supposed  to  have  a  uniform  level  cross  sec- 
tion throughout.  These  tables  are  chiefly  useful  in  making  up 
preliminary  estimates  from  the  profile,  or  in  other  cases  where 
only  approximate  results  are  required.  For  monthly  and  final 
estimates  more  elaborate  tables  are  required,  such  as  are  des- 
cribed in  §  257. 

To  make  an  approximate  estimate  of  quanti- 
ties from  a  profile  by  use  of  Table  XXX. — Select  the 
proper  column  for  base  and  slopes,  and  if  the  outline  of  a  cut 
on  the  profile  is  roughly  a  four-sided  figure,  stretch  a  fine  silk 
thread  over  the  surface  line  to  average  it,  note  the  depth  from 
thread  to  grade  line  midway  of  the  cutting,  and  multiply  the 
tabular  number  opposite  this  depth  by  the  average  length  of 
the  cutting  in  stations  of  100  feet.  (By  average  length  is  meant 
the  half  sum  of  the  length  of  the  grade  line  in  the  cutting  and 
of  so  much  of  the  surface  line  as  is  covered  by  the  thread.)  If 
the  area  of  a  cutting  as  seen  on  the  profile  is  approximately 
triangular,  stretch  an  averaging  line  over  each  incline,  and 
note  the  depth  from  the  intersection  of  these  lines  to  grade, 
and  multiply  the  tabular  number  opposite  this  depth  by  one- 


EXPLANATION   OF  TABLES.  267 

half  the  length  of  the  cut  measured  on  the  grade  line  in  sta- 
tions. The  resulting  quantities  will  be  slightly  in  excess  if  the 
ground  is  level  transversely,  but  may  be  too  small  if  the  trans- 
verse slope  is  steep,  and  cutting  on  the  centre  line  is  small. 
In  general  they  furnish  a  good  approximation.  Quantities  in 
embankments  may,  of  course,  be  found  similarly.  A  cut  or 
fill  may  be  divided  on  the  profile  into  several  portions,  and  the 
contents  of  each  portion  found  separately  if  preferred. 

The  content  of  a  prismoid,  level  transversely,  but  having 
different  end  depths,  maybe  found  correctly  by  this  table  thus: 
add  together  the  quantities  opposite  each  end-depth  and  4  times 
the  quantity  opposite  the  half  sum  of  the  depths;  multiply  the 
sum  by  the  length  in  feet,  and  divide  by  600. 

TABLE  XXXI. — Contains  a  variety  of  useful  numbers  and 
formulae.  The  logarithms  are  here  given  to  seven  places  of 
decimals. 


TABLES. 


269 


TABLE  I.— GEOMETRICAL  PROPOSITIONS. 


The  References  are  to  Davies*  Legendre,  Revised  Edition. 

No. 

REFERENCE. 

HYPOTHESES. 

i 
CONSEQUENCES. 

5      i 

i 

IV.,  XI  

If  a  triangle  is  right 
angled, 

The  square  on  the  hypothe- 
nuse  is  equal  to  the  sum  of 
the  squares  on   the    other 
two  sides. 

2 
3 

L,  XL,  Cor.  1.... 
I     XI 

If  a  triangle  is 
equilateral, 

If  a  triangle  is 
isosceles, 

It  is  equiangular. 

The  angles  at  the  base  are 
equal. 

4 

L,  XL,  Cor.  2.... 

If  a  straight  line 
from  the  vertex 
of  an  isosceles 
triangle  bisects 
the  base, 

It  bisects  the  vertical  angle. 
And  is  perpendicular  to  the 
base. 

5 

I.,  XXV,,  Cor.  6.. 

If  one  side  of  a  tri- 
angle is  pro- 
duced, 

The  exterior  angle  is  equal  to 
the  sum  of  the  two  interior 
and  opposite  angles. 

6 

IV.,  XX  

If  two  triangles  are 
mutually  equian- 
gular, 

They  are  similar.    And  their 
corresponding     sides     are 
proportional. 

7 

L,  XXVII  

If  the  sides  of  a 
polygon  are"  pro- 
duced in  the 
same  order, 

The     sum    of    the    exterior 
angles    equals    four    right 
angles. 

8 
9 

L,  XXVL,  Cor.  1. 
L,  XXVIII  

If  a  figure  is  a 
quadrilateral, 

If  a  figure  is  a 
parallelogram, 

The  sum  of  the  interior  angles 
equals  four  right  angles. 

The  opposite  sides  are  equal. 
The   opposite     angles    are 
equal.    It  is  bisected  by  its 
diagonal.  And  its  diagonals 
bisect  each  other. 

L,  XXXI. 

10 

III.,  VII  

If  three  points  are 
not  in  the  same 
straight  line, 

A    circle     may     be    passed 
through  them. 

11 

in.,  xvn  

If  two  arcs  are  in- 
tercepted on  the 
same  circle, 

They  are  proportional  to  the 
corresponding  angles  at  the 
centre. 

12 
13 
14 

15 

VM  XIII.,  Cor.  2.. 

If  two  arcs  are 
similar, 

If  two  areas  are 
circles, 

If  a  radius  is  per- 
pendicular to  a 
chord, 

If  a  straight  line  is 
tangent  to  a 
circle, 

They  are  proportional  to  their 
radii. 

They  are  proportional  to  the 
squares  on  their  radii. 

It  bisects  the  chord.    And  it 
bisects  the    arc  subtended 
by  the  chord. 

It  meets  it  in  only  one  point. 
And  it  is  perpendicular  to 
the  radius  drawn    to  that 
point. 

III.,  VI  

Ill    IX 

16 

1 

in.,  xrv.,  cor.  .  . 

If  from  a  point 
without  a  circle 
tangents  are 
drawn  to  touch 
the  circle, 

There  are  but  two.    They  are 
equal.       And    they    make 
equal  angles  with  the  chord 
joining  the  tangent  points. 

271 


TABLE  I.— GEOMETRICAL  PROPOSITIONS. 


The  References  are  to  Davies'  Legendre,  Revised  Edition. 

No. 

REFERENCE. 

HYPOTHESES. 

CONSEQUENCES. 

17 

m.,x  

If  two  lines  are 
parallel  chords 
or  a  tangent  and 
parallel  chord, 

They  intercept  equal  arcs  of 
a  circle. 

18 

III.,  XVIH  

If  an  angle  at  the 
circumference  of 
a  circle  is  sub- 
tended by  the 
same  arc  as  an 
angle  at  the  cen- 
tre, 

The  angle  at  the  circumfer- 
ence is  equal  to  half   the 
angle  at  the  centre. 

19 

in.,xvm.,cor.3 

If  an  angle  is  in- 
scribed in  a  semi- 
circle, 

It  is  a  right  angle. 

20 

in.,  xxi  

If  an  angle  is 
formed  by  a  tan- 
gent and  chord, 

It  is  measured  by  one  half  of 
the  intercepted  arc. 

21 

IV.,XXVin.,Cor. 

If  two  chords,  in- 
tersect each  oth- 
er in  a  circle, 

The    rectangle   of    the    seg- 
ments of  the  one,  equals  the 
rectangle  of  the  segments 
of  the  other. 

22 

IV.,XXin.,Cor.2 

And  if  one  chord  is 
a  diameter,  and 
the  other  per- 
pendicular to  it, 

« 

The    rectangle   of    the    seg- 
ments of  the   diameter   is 
equal  to  the  square  on  half 
the  other  chord.    And  the 
half  chord  is  a  mean  pro- 
portional between  the  seg- 
ments of  the  diameter. 

23 

IV.,  XXIX.,  Cor.. 

If  two  secants 
meet  without  the 
circle, 

The  rectangles  of  each  secant 
and    its   external   segment 
are  equal. 

24 

25 

IV.,  XXX.  
IV.,  XIV  

If  a  secant  and 
tangent  meet, 

If  a  straight  line 
from  the  vertex 
of  a  triangle  bi- 
sects its  base, 

The  rectangle  of  the  secant 
and  its  external  segment  is 
equal  to  the  square  on  the 
tangent.     And  the  tangent 
is  a  mean  proportional  be- 
tween the   secant   and   its 
external  segment. 

The  sum  of  the   squares  on 
the  two  sides  is   equal  to 
twice  the  square  of  half  the   \ 
base  increased  by  twice  the 
square  of  the  bisecting  line. 

26 

IV.,  XII  

If  a  perpendicular 
be  drawn  from 
the  vertex  of  a 
triangle  to  the 
base, 

The  square  of  a  side  opposite 
an  acute  angle  is  equal  to 
the  sum  of  the  squares  of 
the  other  side  and  the  base, 
diminished     by    twice   the 
rectangle  of  the  base  and 
the  distance  from  the  ver- 
tex of  the  acute  angle  to 
the  foot  of  the  perpendicu- 
lar. 

272 


TABLE  II.-TRIGONOMETRIC  FORMULA. 


TRIGONOMETRIC  FUNCTIONS. 

Let  A  (Fig.  107)  =  angle  BAG  =  arc  BF,  and  let  the  radius  AF  ~  AB  = 

H=\. 

We  then  have 


sin  .4 
cos  A 
tan  A 
cot  A 
sec  A 
cosec 


=  AG 
=  DF 
=  HG 
-AD 
=  AG 


versin  .4      =  CF  = 
covers  .4    =  j?A'  = 
exsec  A      =  ££> 
coexsec  -4  =  .Bo? 
chord  ^4      =  BF 
chord  2  ^  =  51  = 


In  the  right-angled  ti-iangle  ABC  (Fig.  107) 
Let  AB  =  c,AC  =  b,  and  BC  =  a. 
We  then  have  : 


FIG.  107. 


1.  sin  A 

2.  cos  A 

3.  tan  A 

4.  cot  A 

5.  sec  A 


—     —     =  cos  B 
c 


=  sin  5 

c 


—     =  tan  B 
a 

c 

-7-     =  cosec  B 


cosec  A     =     — 
a 


sec.S 


c  -  6 
7.    vers  J.       =  —   —  =  covers  B 


8.  exsec  A    =  — , —  =  coexsec  # 

o 

9.  covers  A   =  — —  =  versin  B 


11.  a  =  c  sin  yl  =  6  tan  A 

12.  •&  =  c  cos  .A  =  a  cot  A 


sin  ^4       cos  A 


14.  a  =  c  cos  B  =  6  cot  B 

15.  6  —  c  sin  B  =  « tan  5 


ir  __ 

B    cos  B  ~  sin  B 


17.  rt  = 

18.  6  = 

19.  c  = 


(c-  a) 


v  a2  +  6» 


20.      C  =  90°  =  A 


21.  area  =       - 


273 


TABLE  II. -TRIGONOMETRIC  FORMULAE. 


SOLUTION  OF  OBLIQUE  TRIANGLES. 


Fia.  108. 


GIVEN. 

SOUGHT. 

FORMULA. 

22 

A,  B,a 

C,  6,  c 

sin  ad, 

f             ^        •cin  (  A    1     R"» 

-  sin^SmU 

23 

A,a,b 

B,C,c 

sin  5  =  S1^—  .6,             C  =  180°  -  (A  -f  B), 

24 

C,a,b 

»<*•*> 

y*  (A  -f  B)  =  90°  -  %  c 

25 

MU-» 

ran  ^  (A      &)  —             tan  ^  (A  -f-  ^; 

26 

A,  B 

B  =  %(A-}-  B)  —  }&(A  —  B) 

27 

c 

.,   cos  l£(A-{-B)      ,        ,    sin  ^(^4  +  -P) 

28 

area 

^-  =  1^  a  6  sin  C. 

29 

a,  6,  c 

A 

^.^(a  +  .+O^in^y^^c, 

30 

co^^/^^.tan^^y^^ 

.     ^      2Vs(s  -  a)  (s—  6)(s  —  c). 

32 

area 

JT  =  Vs  (s  -  a)  (s  -  6)  (s  -  c) 

33 

A,  B,  C,  a 

area 

g.  _  a2  sin  S  .  sin  C 

274 


TABLE  II.-TRIGOXOMETRIC  FORMULAE. 


GENERAL  FORMULAE. 


sin  A    —     -  .     =     V  1  —  cosa  A     =    tan  A  cos  A 

cosecA 


J4  vers  2  ^4     = 


=    2  sin  J4  A  cos  J^  A    = 
36 
87 


cos  .4    =     -        -     =     4/  f  —  sin2  A     =    cot  ^4  sin  A 
sec  ^ 


cos  A    =     1  -  vers  .4    =     2  cos2  y^A  —  1    =    1—2  sin2  J^  JL 
cos  ^4    =     cos3  J4  ^  —  sin2  ^  A     = 


1  sin  A 

tan  A    —    -        T  ,  SBB    -        ,     =     */  sec2  ^L  —  1 
cot  A  cos  ^1 


45 


46 


48 


50 


tan  J.    =    i-/ — 5-3 1     = 


y    cos2  X    "  cos^4  =    1  4- cos  2 .4 

1  —  cos  2  A  vers  2  ^4 

tan  A     =     -—  .  =  — ; — -     -     =     exsec  A  cot  3^  -4 

sin  2  A  sin  2  ^4 


1  cos  .4 


sjn2^4  sin  2  ^4  1  +  cos  2  -^ 

:    1— ~cos2^    :       vers  2  J.  sin  2  A 


vers  J.     =     1  —  cos  A    =    sin  A  tan  J<j  A    —    2  sin2  ^  ^4 
vers  ^4    =    exsec  A  cos  Jl 


exsec  A    —    sec  ^.  —  1 


sin  2  .,4     =     2  sin  ^4  cos  ^4 


/f+ 

A/  -^- 


cos  2A    =    2  cos2  ^4  —  1    =     cos2  A  —  sin2  A    =    1—2  sin" 
275 


TABLE  II.— TRIGONOMETRIC  FORMULA. 


GENERAL  FORMULA. 


54.  tan  2  A  =      2tan 


o5.  cot.  Mi  A- 


1  —  tan2  A 
sin  A 


J^  vers  ^.  1  —  cos  A 

-  -  =  — 
Vl  —  i^  vers  ^      2  +  Vg  (i  -f-  cos  .4) 


58.  vers  2  A  =  2  sin2  A 

1  —  cos  A 


59.  exsec  ^  A  =  - 


3.  exsec  2  A  = 


(1  +  cos  -4)  +  Va  (l  4-  cos  A) 
tan2  ^4 


1  —  tan2  A 

61.  sin  U  ±  #)  =  sin  ^1 .  cos  B  ±  sin  5 .  cos  A 

62.  cos  (A  ±  B)  =  cos  .4  .  cos  5  T  sin  A .  sin  5 

63.  sin  A-\-sinB  -2sin^(A  +  B)  cos  }£(A  —  B) 
G4.  sin  ^1  —  sin  B  —  2  cos  y%  (A  -J-  /?)  sin  ^(A  —  B) 

65.  cos  ,4  +  cos  J!?  =  2  cos  fc&  04  +  S)  cos  }4  U  -  5) 

66.  cos  B  —  cos  A  =  2  sin  ^  (A  -f  J?)  sin  J^  04  —  B) 

67.  sin2  A  —  sin2  B  =  cos2  B  —  cos2  .4  =  sin  (A  +  5)  sin  (A  —  B) 

68.  cos2  A  —  sin2  #  =  cos  (A  +  JB)  cos  (A  —  B) 


.  cos  B 

276 


TABLE  III.-CURVE  FORMULAE. 


—  — 

GIVEN.                    SOUGHT. 

FORMULA. 

D 

J? 

J2  =  sln^ 

2 

R 

» 

sm^Z>=  -g- 

3 

A,   D 

£ 

i^lOO-^ 

4 

D,  L 

A 

Di 

5 

A,  L 

D 

J>  =  100^- 

6 

R,    A 

r 

T  =  R  tan  ^  A 

T 

*» 

c 

C  =  2  J?  sin  ^  A 

8 

M 

M 

M=  R  vers  ^  A 

a 

M 

E 

E  =  R  exsec  J^  A 

10 

Tr    A 

R 

.R=  Tcot^  A 

ii 

"• 

E 

E  =  T  tan  J4  A 

12 

M 

C 

C  =  2  T  cos  ^  A 

IS 
14 

E,    A 

M 

R 

Jf  =  T  cot  1^  A  .  vers  J^  A 

K  ~  exsec  J^  A 

15 

" 

T 

T  =  E  cot  J4  A 

IS 

" 

C 

exsec  Jx£  A 

17 

« 

M 

Jf=#COS^A 

18 

C,    A 

R 

jB  =  !Tsin^~A" 

19 

20 

« 

M 
T 

Jf  =  U  C  tan  M  A 
T  —             ^ 

2  COS  ^  A 

21 

r 

E 

F  —  u  r»  exsec  ^  A 

22 

Jf,    A 

R 

*.—  ^  * 

vers  J^  A 

23 

« 

C 

C  =  2  M  cot  M  A 

24 
25 

K 

T 
E 

r  _    Jf  tan  ^  A 

~    COS  J^  A 

277 


TABLE  III.-CURVE  FORMULAE. 


26 

GIVEN.             SOUGHT. 

FORMULAE. 

R,   T 

A 

tan  14  A  =  —  — 

27 

28 
29 

R,  C 
M 

A 

T 

sin  J^  A  =  —  — 

COS^A=,     1|/(B  +  -|)(S-|-) 

30 

R,  M 

Jf 

vers  J^  A  =  —  — 

31 

" 

" 

COS  J^  A    =  ^  

32 

R,  E 

A 

exsec  J^  A  =  —  p- 

33 

«• 

" 

cosK3A=ir^ 

34 
35 

T,  c 

u 

A 

COS  *  A    =  -^ 

tan  y±  A  =  |/l|^g 

36 

T,E 

A 

tan  M  A  =  ^~ 

37 

if 

« 

COS  J^  A    =  y-aT    ^;a 

38 
39 

C1    Hf 

-.: 

A 

;•  J$ 

23f 

cos  ^  A  =  -Qf-f+tfi 

40 
41 
42 
43 
44 

Jf  ,  £ 
R,'T 

A 

3/ 

. 

« 

COS  ^   A    =  -=- 
jE/ 

_                  6  J.  ±\f 

M  —  R 

E  =   V  21  2  -f-  -K2  —  -B 

45 

46 

47 

R,  C 
tt 

r 
.if 

.E 

r_                         CK 

y(*+i)(«-l) 

J?2 

=  v(«HF*c?H«~X  C)  ~  R 

278 


TABLE  III. -CURVE  FORMULAE. 


GIVEN. 

SOUGHT. 

FORMULAE. 

48 
49 

A  M 

T 
C 

*                        _                '          

T  —  —                     —  -  - 

C  =    2VM(2R  -  M) 

50 
51 

R,  E 

E 
T 

E  =  ^~^~M 

T  =  VE&R  +  E) 

52 
53 
54 

T,  C 

C 
M 
R 

f^,  _  2  R  /j/  E  (2  R  4  E) 

B  +  E 
CT 

~   \'(2T+C)  (2T-C) 

55 

" 

M 

M=y*c\/lTT~+cc- 

56 

" 

E 

E=T\/'2T^-        - 

57 

T,  E 

r> 

R=I    ^Jp 

58 

« 

c 

2  T  (T2  —  jE12) 

~~  T^'+E*  —  ' 

59 

" 

M 

M=~T*+E*- 

60 

C,  M 

R 

E<=-M"'\-M~ 

61 
62 
63 

M,  E 

T 
E 
R 

C  (C  2  4-  4  M  2) 
2(C^-T~.M2) 

L^:4"2  ,;  5 

64 

« 

T 

r=£i/!±:« 

65 

« 

C 

c=^V:^-^ 

66 

T,  M 

R 

2  J/ 

67 
68 

«    . 

E 

C 

E  3  +  £  2  M  —  ET  2  4-  MT  ~  =  0 
C'3  4  2  TC2  4  4  M2  C  -  8  Jf  2  T  _  o 

69 

C,  E 

R            v  +  v^ljjLCL  -B  ~  -    C1E  =  0 

70 

" 

T 

g  rs  _  ra  c—  2  T^2  -  C£7*  =  0 

71 

M 

4              4 

TABLE  IV.— RADII,  LOGARITHMS,  OFFSETS,  ETC. 


f- 

'   Dog. 

Radius. 

Loga- 
rithm. 

Tan. 
Off. 

Mid. 
Ord. 

[ 
Deg. 

Radius. 

Loga- 
rithm. 

Tang. 
Off 

Mid. 
Ord. 

I>. 

11. 

log.  K. 

t. 

in. 

D. 

R. 

log.  K. 

t. 

in. 

0°  o' 

Infinite 

Infinite 

.000 

.000 

1°    0' 

5729.65 

3.758128 

.873 

.218 

i 

343775. 

5  530274 

.015 

.001 

1 

5(535.72 

.750950 

.887 

.222 

2 

171887. 

5  235244 

.029 

.007 

2 

5544.83 

.743888 

.902 

.225 

3 

114592. 

5.059153 

.044 

.011 

3 

5456.82 

.736939 

.916 

.229 

4 

85943.7 

4.934214 

.058 

.015 

4 

5371.56 

.730100 

.931 

.233 

5 

68754.9 

.837304 

.073 

.018 

5 

5288.92 

.723367 

.945 

.236 

6 

57295.8 

.758123 

.087 

.022 

6 

5208.79 

.716737 

.960 

.240 

7 

49110.7 

.691176 

.102 

.025 

7 

5131.05 

.710206 

.974 

.244 

8 

42971.8 

.633184 

.116 

.023 

8 

5055.59 

.703772 

.939 

.247 

9 

38197.2 

.532031 

.131 

..033 

9 

4982.33 

.697432 

1.004 

.251 

10 

34377.5 

4.536274 

.145 

.036 

10 

4911.15 

3.691183 

1.018 

.255 

11 

3125-3.3 

4.494831 

.160 

.040 

11 

4841.98 

3.685023 

1.033 

.258 

12 

23347.8 

.457o:H 

.175 

.044 

12 

4774.74 

.678949 

1.047 

.262 

13 

23441.2 

.4323*1 

.189 

.047, 

13 

4709.33 

.67'2959 

.062 

.265 

14 

24.yr>.4 

.330146 

.204 

.051 

14 

4645.69 

.667051 

.076 

.269 

15 

2291H.3 

.360183 

.218 

.055 

15 

4583.75 

.661221 

.091 

.273 

16 

21H5.9 

.332154 

.2-33 

.658 

16 

4523.44 

.655469 

.105 

.276 

17 

2)233.1 

.305825 

.247 

.032 

17 

4464.70 

.649792 

.120 

.230 

18 

19033.0 

.231003 

.232 

.065 

18 

4407.46 

.644189 

.134 

.284 

19 

18333.4 

.257521 

.276 

.069 

19 

4:351.67 

.638656 

.149 

.287 

20 

17188.8 

4.235244 

.291 

.073 

23 

4297.28 

3.6=33194 

.164 

.291 

21 

16370.2 

4.214055 

.305 

.076 

21 

4244.23 

3.627799 

.178 

.295 

22 

15626.1 

.193852 

.320 

.0:30 

22 

4192.47 

.622470 

.193 

.298 

23 

14:>46.7 

.174547 

.335 

.034 

23 

4141.96 

.6172J6 

.207 

.302 

24 

143-33.0 

.156034 

.349 

.087 

24 

4092.66 

.612005 

222 

.305 

25 

13751.0 

.138335 

.364 

.091 

25 

4044.51 

.606866 

iaae 

.309 

26 

13-333.1 

.121302 

.378 

.095 

26 

3997.49 

.601787 

.251 

.313 

27 

13732.4 

.104911 

.393 

.093  1 

27 

3951.54 

.596766 

.235 

.316 

28 

12377.7 

.089117 

.407 

.102  1 

23 

3906.54 

.591803 

.230 

.320 

29 

11854.3 

.073877 

.422 

.105 

29 

3862.74 

.586396 

1.294 

.324 

30 

11459.2 

4.059154 

.436 

.109 

30 

3819.83 

3.582044 

1.309 

.327 

31 

11039.6 

4.044914 

.451 

.113 

•31 

3777.85 

3.577245 

1.324 

.331 

32 

10743.0 

.031125 

.465 

.116 

32 

3736.79 

.572499 

1.338 

.335 

33 

10417.5 

.017762 

.480 

.120 

33 

3396.61 

.567804 

1.353 

.338 

34 

10U1.1 

4.004797 

.495 

.124 

34 

3657.29 

.563160 

1.367 

.342 

35 

9i33.l3 

3.998203 

.509 

.127 

&5 

3318.80 

.5585(54 

1.382 

.345 

36 

9.549.34 

.979973 

.524 

.131 

36 

,3581.10 

.554017 

1.396 

.349 

37 

9331.23 

.938074 

.538 

.135 

37 

3544.19 

.549517 

1.411 

.353 

38 

904(5.75 

.956493 

.553 

.138 

38 

3508.02 

.545063 

1.425 

.356 

39 

8S14.7-J 

.945212 

.567 

.142 

39 

3472-59 

.540354 

1.440 

.360 

40 

8.594.42 

3.934216 

.58* 

.145 

40 

3437.87 

3.53G283 

1.454 

.3(34 

41 

8384.80 

3.923493 

.596 

.149 

41 

3403.83 

3.531963 

1.469 

.367 

42 

8185.16 

.913027 

.(511 

.153 

42 

3370.46 

.527690 

1.483 

.371 

43 

7934.81 

.932803 

.635 

.156 

43 

8-337.74 

.523453 

.493 

.375 

44 

7813.11 

.892824 

.640 

.160 

44 

3305.65 

.519257 

.513 

.378 

10 

7iB-).4'.) 

.8-53065 

.654 

.164 

45 

3-374.17 

.515101 

.527 

.382 

41) 

7473.4-3 

.873519 

.669 

167 

46 

3243.29 

.510985 

.542 

.385 

47 

7314.41 

.864179 

.684 

.171 

47 

3212.93 

.503908 

.556 

.389 

48 

710-3.05 

.855036 

.(593 

.174 

48 

3183.23 

.5Q3868 

.571 

.393 

49 

ri)ir,.s7 

.846382 

.713 

.178 

49 

3154.03 

.498866 

.585 

.396 

50 

6875.55 

3.837303 

.727 

.182 

50 

3125.33 

3.494900 

.600 

.400 

51 

6740.74 

3.823703 

.742 

.185 

51 

3097.20 

3.490970 

.614 

.404 

52 

0611.12 

.820275 

.756 

.189 

52 

3069.55 

.487075 

.629 

.407 

53 

6483.33 

.812002 

.771 

.193 

53 

3042.39 

.483215 

.643 

.411 

-  54 

6366.2(5 

.803885 

.785 

.196 

54 

3315.71 

.479389 

.658 

.414 

55 

(J-3.50.5l 

.795916 

.800 

.200 

55 

2939.43 

.  475596 

.673 

.418 

56 

6138.93 

.783091 

.814 

.204 

56 

2933.71 

.471836 

.687 

.422 

57 

6031.23 

.780404 

.829 

.207 

57 

2338.39 

.468109 

.702 

.425 

58 

5927.22 

.772851 

.844 

.211 

58 

2913.49 

.464413 

.716 

.429 

59 

5826.76 

.765427 

.858 

.215 

59 

2389.01 

.460749 

1.731 

.433 

i         60 

5729.65 

3.758128 

.873 

.218 

60 

2364.93 

3.457115 

1.745 

.436 

280 


TABLE  IV.— RADII,  LOGARITHMS,  OFFSETS,  ETC. 


Deg. 
D. 

Radius. 
K. 

Loga- 
riuim. 

log.  K. 

TSg' 

t. 

Mid.  ! 
Orel. 
m.    | 

Deg. 
». 

Radius. 
B. 

Loga- 
rithm. 

log.  R. 

'S3?- 

t. 

M;«!. 

Ord. 
in. 

2°  0' 

2864  93 

3  457115 

1.745 

.436 

3°  0' 

1910.08 

3.281051 

2.618 

.054 

i 

2841  :  26 

.453511 

1.760 

.440  1 

1899.53 

.278646 

2.632 

ioES 

2 

2817.97 

.44UU87 

1  .  774 

.444  -i 

2 

1889.09 

.276253 

2.647 

.002 

3 

2795.06 

.440392 

.789 

.447 

3 

1878.77 

.273874 

2.601 

.665 

4 

2i  72.  53 

.442876 

.803 

.451 

4 

1868.56 

.2710)8 

2.076 

.009 

5 

2750.85 

.43U&8 

.818 

.454 

5 

1858.47 

.269155 

2.600 

.073 

6 

2728.52 

.4359x8 

.832 

.458 

6 

1848.48 

.266814 

2.705 

.676 

2707.04 

.432495 

.847 

.462 

7 

1838.59 

.264486 

2.719 

.080 

8 

2685.89 

.429089 

,862 

.465  ! 

8 

1828.82 

.262170 

2.734 

.084 

9 

2665.08 

.425710 

.876 

.469 

9 

1819.14 

.259867 

2.749 

.687 

10 

2644.58 

3  422356 

.891 

.473 

10 

1809.57 

3.257576 

2.763 

.691 

11 

2624.39 

3.419029 

.£05 

.476 

11 

1800.10 

3.255296 

2.778 

.694 

1:3 

2604.51 

.415727 

.920 

.480 

12 

1790.73 

.253029 

2.792 

.698 

13 

2584.93 

.412449 

.934 

.484 

•     13 

1781.45 

.250774 

2.807 

.702 

14 

2565.65 

.409197 

.949 

.487 

14 

1772.27 

.248530 

2  821 

.705 

15 

2546.  64 

.405968 

.963 

.491 

15 

1763.18 

.246297 

2.836 

.7'09 

10 

2527.92 

.402763 

.978 

.494 

16 

1754.19 

.244077 

2.850 

.713 

17 

2509.47 

.399582 

.992 

.498 

17 

1745.26 

.241867 

2.805 

.716 

18 

2491.29 

.396424 

2.007 

.502 

18 

1736.48 

.239669 

2.87'9 

.720 

19 

2473.37 

.393289 

2.022 

.505 

19 

1727.75 

.237481 

2.894 

.723 

20 

2455.70 

3  390176 

2.036 

.509 

20 

1719.12 

8.235305 

2.908 

.727 

21 

2438.29 

3.887085 

2.051 

.513 

21 

1710.56 

3.233140 

2.923 

.731 

22 

2421.12 

.384016 

2.065 

.516 

22 

1702.10 

.230985 

2.988 

.734 

23 

2404.19 

.380969 

2.080 

.520 

23 

1693.72 

.228841 

2.952 

.738 

24 

2387.50 

.377943 

2.094 

.524 

24 

16a5.42 

.226707 

2.967 

.742 

26 

2371.04 

.374938 

2.109 

.527 

25 

1677.20 

.224584 

2.981 

.745 

26 

2354.  KJ 

.371954 

2.123 

.531 

26 

1669.06 

.222472 

2.996 

7'49 

27 

2&S8.78 

.368990 

2.138 

.534 

27 

1661.00 

.220369 

3.010 

.753 

28 

2322.98 

.366046 

2.152 

.538 

28 

1653.01 

.218277 

8.025 

.756 

29 

2307.39 

.363122 

2.167 

.542 

29 

1645.11 

.216195 

3.039 

.700 

30 

2292.01 

3.360217 

2.181 

.545 

30 

1637.28 

3.214122 

3.054 

.763 

31 

2276.84 

3.357332 

2.196 

.549 

31 

1629.52 

3.212060 

3.  008 

.767 

32 

2261.86 

.&54466 

2.211 

.553 

32 

1621.84 

.210007 

3.083 

.  771 

33 

2247.08 

.351618 

2.225 

.556 

33 

"1614.22 

.207964 

3.097 

.774 

34 

2232.49 

.348789 

2.240 

.560 

34 

1606.68 

.205930 

3.112 

.778 

35 

2218.09 

.345979 

2.254 

.564 

35 

1599.21 

.203906 

3.127 

.782 

36 

2203.87 

.343187 

2.269 

.567 

36 

1591.81 

.201892 

3.141 

.785 

37 

2189.84 

.340412 

2.283 

.571 

37 

1584.48 

.199886 

3.1.-0 

.789 

38 

2175.98 

.337655 

2.298 

.574 

38 

1577.21 

.197890 

3.170 

.793 

39 

2162.30 

.334916 

2.312 

.578  , 

39 

1570.01 

.195903 

a!  185 

.  7'IHJ 

40 

2148.79 

3.332193 

2.327 

.582' 

40 

1562.88 

3.1«m» 

3.19!) 

.800 

41 

2135.44 

3.329488 

2.341 

.585 

41 

1555.81 

3.191950 

3.214 

.803 

42 

2122.26 

.326799 

2.356 

.589 

42 

1548.80 

.189:)% 

3.228 

.807 

43 

2109.24 

.324127 

2  371 

.593  ; 

43 

1541.86 

.188045 

3.243 

.811 

44 

2096.  39 

.321471 

2.385 

.596 

44 

1584.98 

.18(3103 

3.157 

.814 

45 

2083.68 

.318832 

2.400 

.600  ! 

45 

1528.16 

.1K41C9 

3.272 

.818 

46 

2071.13 

.316208 

2.414 

.604 

46 

1521.40 

.182244 

3.2S6 

.822 

47 

2058.73 

.313600 

2.429 

.607  ! 

47 

1514.70 

.1808S7 

3.301 

.825 

48 

2046.48 

.311008 

2.443 

.611  i 

48 

1508.06 

.178419 

3.316 

.829 

49 

2034  37 

.308431 

2.458 

.614 

49 

1501.48 

.176519 

8.830 

.832 

50 

2022.41 

3.305869 

2.47'2 

.618 

50 

1494.95 

3.174027 

3.345 

.830 

51 

2010.59 

3.30,3323 

2  487 

.622 

51 

1488.48 

3.17S744 

3.  £39 

.840 

52 

1998.90 

.300791 

2.501 

.625 

52 

1482.07 

.170868 

3.374 

.843 

53 

1987.35 

.298274 

2.516 

.629 

53 

1475.71 

.169001 

3.388 

.847 

54 

1975.93 

.295771 

2-530 

.633 

54 

1469.41 

.167142 

3.403 

,651 

55 

1964.64 

.293283 

2.545 

.636 

55 

1463.16 

.165291 

3.417 

.854 

56 

19:53.48 

.290809 

2.560 

.640 

56 

1456.96 

.163447 

3  .  m 

.858 

57 

1942.44 

.288349 

2.574 

.644 

57 

1450.81 

.161612 

3.446 

.802 

58 

1931.53 

.285902 

2  589 

.647 

58 

1444.72 

.159784 

3.461 

.865 

59 

1920  75 

.283470 

2.603 

.651 

59 

1438.68 

.157963 

3.475 

.809 

60 

1910.08 

3.281051 

2.618 

.654 

60 

1432.69 

3.156151 

3.490 

.872 

281 


TABLE  IV.— RADII,  LOGARITHMS,  OFFSETS,  ETC. 


Deg. 
D. 

"•Hiss: 

R.    log.  K. 

Tang. 

oc 
t. 

Mid. 
Ord. 

111. 

Deg. 
D. 

*•*»•  HtS 
K,   !  log.R. 

"*& 

t. 

Mid. 
Ord. 

in. 

4°  0'  1433.09  3.156151  3.490 

.872 

5°  0'  1146.28 

3.059290 

4.362 

1.091 

1   1426.74   .154346  3.505 

.876 

1  I  1142.47 

.057846 

4.376 

1.094 

2  !  1420.85  •  .152548  3.519 

.880 

2  i  1138.69 

.056407 

4.391 

1.098 

3  i  1415.01  i  .150758  3.534 

.883 

3  i  1134.94 

.054972 

4.405 

1.102  1 

4 

1409.21  I  .148975 

3.548 

.887 

4 

1131.21   .053542 

4.420 

1.105  ! 

5 

1403.46   .147200 

3.503 

.891 

5 

1127.50 

.052116 

4.435 

1.109 

6 

1397.76   .145431 

3.577 

.894 

6 

1123.82 

.050696 

4.449 

1.112 

7 

1392.10  j  .143670 

3.592 

.898 

7 

1120.16 

.049280 

4.464 

1.116 

8 

138(5.49   .141916  j  3.606 

.902 

8 

1116.52 

.047868 

4.478 

1.120 

9 

1380.92  i  .140170  3.621 

.905 

9 

1112.91 

.046462 

4.493 

1.123 

10  i  1375.40  i3.138430  3.635 

.909 

10 

1109.33  |3.045059 

4.507 

1.127 

11   1369.92  3.136697  3.650 

.912 

11 

1105.76  3.043662 

4.522 

1.181 

12  1364.49   .134971   3.664 

.916 

12 

1102.22   .042268 

4.536 

1.134 

13  1359.10   .133251   3.679 

.920 

13 

1098.70 

.040880 

4.551 

1.138 

14  1353.75   .131539  1  3.693 

.923 

14 

1095.20 

.039495 

4.565 

1.142 

15  1348.45   .1298:33 

3.708 

.927 

15  1091.73   .038115 

4.580 

1.146 

16  1343.15   .128134 

3.723 

.931 

16 

1088.28 

.036740 

4.594 

1.149 

17  1337.65 

.120442  3.736 

.934 

17 

1084.  85 

.035808 

4.609 

1.153 

18  |  1332.77 

.124756 

3.752 

.938 

18 

1081.44 

.034002 

4.623 

1.157 

19 

1327.63 

.123077 

3.766 

.942 

19 

1078.05 

.032639 

4.638 

1.160 

20 

1322.53  |3.121404 

3.781 

.945 

20  1074.68 

3.031281 

4.653 

1.164 

21   1317.46  3.119738 

3.795 

.949 

21  1071.34 

3.029927 

4.667 

1.168 

22 

1312.43 

.118078  3.810 

.952 

22  1068.01 

.028577 

4.682 

1.171 

23 

1307.45 

.116424 

3.824 

.956 

23  1064.71 

.127231 

4.696 

1.175 

24 

1302.50 

.114777 

3.839 

.960 

24  1061.43 

.025890 

4.711 

1.179 

25  1297.58 

.113136 

3.853 

.963 

25 

1058.16 

.024552 

4.725 

1.182 

26  1292.71 

.111501 

3.868 

.967 

26 

1054.92 

.023219 

4.740 

1.186 

27  1287.87 

.109872  3.882 

.971 

27 

1051.70 

.021890 

4.754 

1.190 

28 

1283.07 

.108249  3.897 

.974 

28 

1048.48 

.020505 

4.709 

1.193 

29  |  12;  8.  30 

.100032  3.911 

.978 

29 

1045.31 

.019244 

4.783 

1.197 

30  1273.57 

3.105022  3.926 

.982 

30 

1042.14 

3.017927 

4.798 

1.200 

31 

1268.87 

3.103417  3.941 

.985 

31 

1039.00 

3.016314 

4.812 

1.204 

32 

1264.21 

.101818 

3.955 

.989 

32 

1035.87 

.015305 

4.827 

1.208 

33 

1259.58 

.100225 

3.970 

.993 

33 

1032.76 

.013999 

4.841 

1.211 

34 

1264.08 

.098638 

3.984 

.996 

34 

1029.67 

.012098 

4.856 

1.215 

35 

1250.42 

.097057 

3.999 

1.000 

35 

1026.60 

.011401 

4.870 

1.218 

36 

1245.80 

.095481 

4.013 

1.003 

36 

1023.55 

.010107 

4.885 

1.222 

37 

1241.40 

.093912 

4.028 

1.007 

37 

1020.51 

.008818 

4.900 

1.226 

38 

1236.94 

.092347 

4.042 

1.011 

38 

1017.49 

.00.032 

4.914 

1.229 

39 

1232.51 

.090789 

4.057 

1.014 

39 

1014.50 

.006250 

4.929 

1.233 

40 

1228.11 

3.089236 

4.071 

1.018 

40 

1011.51 

3.004972 

4.943 

1.237 

41 

1223.74 

3.087689 

4.086 

1.022 

41 

1008.55 

3.003698 

4.958 

1.240 

42 

1219.40   .086147 

4.100 

1.025 

42 

1005.60 

.002427 

4.972 

1.244 

43 

1215.30 

.084610 

4.115 

1.029 

43 

1002.67 

3.001160 

4.987 

1.247 

44 

1210.82 

.083079 

4.129 

1.032 

44 

999.762 

2.999897 

5.001 

1.251 

45 

1206.57 

.081553 

4.144 

1.036 

45 

996.867 

.998637 

5.016 

1.255 

46 

1202.36 

.080033 

4.159 

1.040 

46 

993.988 

.997381 

5.030 

1.258 

47 

1198.17 

.078518 

4.173 

1.043 

47 

991.126 

.996129 

5.045 

1.262 

48 

1194.01 

.077008  1  4.188 

1.047 

48 

988.280 

.  994H80 

5.059 

1.266 

49 

1189.88 

.075504 

4.202 

1.051 

49 

985.451 

.993635 

6.074 

1.269 

50 

1185.78 

3.074005 

4.217 

1.054 

50 

982.638 

2.992393 

5.088 

1.273 

51 

1181.71 

3.072511 

4.231 

1.058 

51 

979.840 

2.991155 

5.103 

1.277 

52 

1177.06 

.071022 

4.246 

1.062 

52 

977.060 

.989921 

5.117 

1.280 

53 

1173.65 

.069538 

4.260 

1.065 

53 

974.294 

.988690 

5.132 

1.284 

^  54 

1169.06 

.068059 

4.275 

1.069 

54 

971.544 

.987463 

5.146 

1.288 

55 

1165.70   .066585 

4.289 

1.073 

55 

968.810 

.986238 

5.161 

1.291 

56  i  1161.76 

.065116 

4.304 

1.076 

56 

966.091   .985018 

5,175 

1.295 

57  1157.85 

.063653 

4.318 

1.080 

57  963.387  i  .983801 

5.190 

1.298 

58  1153.97  |  .062194 

4.333 

1.083 

58 

960.698   .982587 

5.205 

1.302 

59 

1150.11  j  .060740 

4.347 

1.088 

59 

958.025 

.981377 

5.219 

1.306 

60 

1146.28  3.059290 

4.362 

1.091 

60 

955.366  2.980170 

5.234 

1.309 

TABLE  IV.— RADII,  LOGARITHMS,   OFFSETS,  ETC. 


Deg. 

Radius. 

Loga- 
rithm. 

Tang. 

Off. 

Mid. 
Ord. 

Deg.  !  Radius. 

Loga-     Tang, 
rithin.       Off. 

Mid. 
Ord. 

I>.           R. 

log.  R.        t. 

in. 

D.           R. 

log.  R.       t. 

m. 

6°  O'j  955.  3G6   2.980170 

5.234  I  1.300 

7°  0'     819.020  12.913235 

6.105 

1.528 

1 

952.7'22 

.978966 

5.248     1.313 

1 

817.077 

.912:363     6.119 

1.531 

2 

950.093 

.977766 

5.263 

1.317 

2 

815.144 

.911234  1  6.134 

1.535 

3 

947.478 

.976569 

5.277 

1.320 

3 

813.238 

.910208 

6.148 

1.539 

4 

944.877 

.975375 

5.292 

1.324 

4 

811.303 

.909183 

6.163 

1.543 

5 

942.291 

974185 

5.306 

1.327 

5 

809.397 

.908162     6.177 

1.546 

6 

939.719 

.972998 

5.321 

1.331 

6 

807.499 

.907142  1  6.192 

1.550 

7 

937.161 

.971814 

5.335 

1.335 

7 

805.611 

.906125  I  6.206 

1.553 

8 

934.616 

.970633 

5.350 

1.338 

8 

803.731 

.  905111      6.221 

1.557 

9 

932.086 

.909456 

5.364 

1.342 

P 

801.860 

.904098      6.236 

1.501 

10 

920.569    2.968282 

5.379 

1.346 

10 

799.997 

2.903089 

6.250 

1.564 

11 

927.066    2.967111 

5.393 

1.349 

11 

798.144 

2.902081 

6.265 

1.508 

12 

924.576 

.965943 

5.408 

1.353 

12 

796.299 

.901076 

6.27-9 

1.57-2 

13 

922.100 

.964778 

5.422 

1.356 

13 

794.462 

.900073      6.294 

1.575 

14 

919.637      .963616  1  5.437     1.360 

14 

792.634 

.899073     6.803 

1.579 

15 

917.187  l    .96245815.451      1.364 

15 

790.814 

.898074 

6.323 

1.582 

16 

914.750 

.961303 

5.466      1.368 

16 

789.003 

^97078     6.337 

1.586 

17 

912.326 

.900150 

5.480 

1.371 

17 

787.210 

.896085 

6.352 

1.590 

18 

909.915 

.959001 

5.495 

1.375 

18 

785.405 

.895094 

6.366 

1.593 

19 

907.517 

.957855 

5.510 

1.378 

19 

783.618 

.894105 

6.381 

1.597 

20 

905.131 

2.956711 

5.524 

1.382 

20 

781.840   2.89^118 

6.395 

1.600 

21 

902.758 

2.955571 

5.539 

1.386 

21 

780.089  :2.  892133 

6.410 

1.604 

22 

900.397 

.954434 

5.553 

1.389 

22 

778.307 

.891151      6.424 

1.608 

23 

898  048 

.953300 

5.568 

1.393 

23 

776.552 

.890171 

6.439 

1.011 

24 

895.712 

.952168 

5.582 

1.397 

24 

774.806 

.889193 

6.453 

1.615 

25 

893,388 

.951040 

5.597 

1.400 

25 

773.067 

.888217     0.408 

1.019 

26 

891.076 

.949915 

5.611 

1.404 

26 

771.386 

.887244 

6.482 

1.023 

27 

888.776 

.948792 

5.626 

1.407 

27 

769.613 

.88627-2 

6.497 

1.026 

28 

886.488 

,94707-3 

5.640 

1.411 

28 

767.897 

.885303 

6.511 

1.6EO 

29 

884.211 

.946556 

5.655 

1.415 

2!) 

706.190 

.884336 

6.526 

1.083 

30 

881.946    2.945442 

5.669 

1.418 

30 

764.489 

2.883371 

6.540 

1.037 

31 

879.693    2.944331 

5.684 

1.422 

31 

762.797 

2.882409 

6.555 

1.041 

32 

877.451      .943223 

5.698 

1.426 

32 

761.112 

.881448     6.569 

1.044 

33 

875.221      .942118 

5.713 

1.429 

33 

759.434 

.880490     6.584 

1.04'  5 

34 

873.002      .941015  !  5.7'27 

1.433 

34 

757.764 

.87'9534      6.598 

1.651 

35 

870.795 

.939916 

5.742 

1.437 

S5 

756.101 

.87'8580  |  6.613 

1.655 

36 

868.598 

.938319 

5.756 

1.440 

88 

754.445 

.877627 

6.627 

1.659 

37 

866.412      .937725      5.771 

1.444 

37 

752.796 

.876678 

6.642 

1.662 

38 

864.238      .936033 

5.785 

1.447 

38 

7'51.155 

.8757-30 

6.656 

1.0C6 

39 

862.075      .935545 

5.800 

1.451 

39 

7'49.521 

.874784 

0.671 

1.070 

40 

859.922    2.034459 

5.814 

1.455 

40 

747.894   2.873840 

6.085 

1.073 

41 

857.780   2/J33376 

5.829     1.458 

41 

746  271 

2.872898 

6.700 

1.677 

42 

855.648      .932295 

5.844      1.462 

42 

744.001 

.871859 

6.714 

1.080 

43 

853.52?      .931218     5.858 

1.466 

43 

743.055 

.871021 

6.729 

1.084 

44 

851.417      .930142      5873 

1.469 

44 

741.456 

.870086 

6.743 

1.688 

45 

849.317      .929070     5.887 

1.473 

45 

739.864 

,869152 

6.758 

1.691 

46 

847.228      .928000     5.902     1.476 

46 

738.279      .868221 

6.773 

1.695 

47 

845.148 

.926933      5.916  j  1.480 

47 

736.701      .867'291 

6.787 

1.699 

48 

843  080      .925809 

5.931 

1.484 

48 

735.12:)      .866363 

6.802 

1.702 

49 

841.021      .924807 

5.945 

1.487 

49 

7'33.  564  !   .865438      6.816 

1.706 

50 

838.972 

2.923747  ;  5.960 

1.491 

50 

732.005  J2.  864514     6.831 

1.710 

51 

836.933 

2.922691      5.974 

1.495 

51 

730.454 

2.863593 

6.845 

1.713 

52 

834.904 

.921637     5.989 

1.498 

52 

798.909 

.862673 

6.860 

1.717 

53 

832.885 

.920585     6.003 

1.502 

53 

727.370 

.8617-55 

6.874 

1.7SQ 

51 

&30.876      .919536      6.018 

1.505 

54 

725.838 

.8608-10 

6.889 

1.724 

55 

828.876 

.918489     6.032 

1.510 

55 

7-24.312 

.859926 

6.903 

1.728 

56 

826.886 

.917446  i  6.047 

1.513 

56 

722.793 

.859014 

6.918 

1.731 

57 

824.905 

.916404      6.061 

1.517 

57     7'21.280 

.858104 

6.932 

1.735 

58 

822.934      .915365     6.076 

1.520 

58 

719.774 

.857196 

6.947 

1.739' 

59 

820.973      .914329      6.090 

1.524 

59 

718.273      .856290  1  6.961 

1.742 

60 

819.020    2.913295     6.105 

1.528 

60 

716.779   2.855385     6.976 

1.746 

TABLE  IV.— RADII,   LOGARITHMS,   OFFSETS,  ETC. 


Deg. 
D. 

Radius. 
K. 

Loga- 
rithm. 

log.  B. 

155? 

t. 

Mid. 
Ord. 

in. 

Deg. 
». 

Radius. 
K. 

Loga- 
rithm. 

log.  K. 

185? 

t. 

Mid. 
Ord. 

in. 

8°  0' 

716.779 

2.855385 

6.976 

1.746 

9°  0' 

C37.275 

2.804327 

7.  846 

1.965 

1 

715.291 

.854483 

6.990 

1.749 

l 

636.099 

.803525 

7.860 

1  968 

2 

713.810 

.853583 

7.005 

1.753 

2 

634.928 

.8027^4 

7.875 

1.972 

3 

712.335 

.852684 

.019 

1.756 

3 

633.761 

.801C26 

7.889 

1.975 

4 

710.865 

.851787 

.034 

1.761 

4 

632.599 

.801128 

7.904 

1.979 

b 

709.402 

.t  50892 

.048 

1.704 

5 

631.440 

.800332 

r.»i8 

1  .  983 

6 

707.945 

.849999 

.063 

1.768 

6 

630.286 

.799538 

7.933 

1.987 

7 

706.493 

.849108 

.077 

1.771 

7 

629.136 

.798745 

7.947 

1.990 

8 

705.048 

.848219 

.092 

1.775 

8 

627.991 

.797953 

7.962 

1.994 

9 

703.609 

.847331 

.106 

.778 

9 

626.849 

.797163 

7  976 

1.998 

10 

702.175 

2.846445 

.121 

.782 

10 

625.712 

2.796374 

7.991 

2.001 

11 

700.748 

2.845562 

.135 

.786 

11 

624.579 

2.795587 

8.005 

2.005 

12 

699.326 

.844679 

.150 

.790 

12 

623.450 

.794801 

8.020 

2.008 

13 

697.910 

.84875)9 

.164 

.793 

13 

622.325 

.794017 

8.034 

2.012 

14 

696.499 

:  842921 

.179 

.797 

14 

621.203 

.793234 

8.049 

2.016 

15 

695.095 

.842044 

.193 

.801 

19 

620.087 

.^92453 

8.063 

2.019 

16 

693.696 

.841169 

.208 

.804 

16 

618.974 

.791673 

0.078 

2.023 

17 

692.302 

.840296 

.222 

.807 

17 

617.865 

.790894 

8.092 

2.026 

18 

690.914 

.839424 

.237 

.811 

18 

616.760 

.790117 

8.107 

2.030 

19 

689.532 

[838555 

.251 

.815 

19 

615.  6CO 

.789341 

8.121 

2.034 

20 

688.156 

2.837687 

.266 

.819 

20 

614.C63 

2.788566 

8.136 

2.037 

21 

686.785 

2.836821 

.280 

.822 

21 

613.470 

2.787793 

8.150 

2.041 

22 

685.419 

.8M5si.50 

.295 

.826 

22 

612.380 

.787021 

8.165 

2.045 

23 

684.059 

.835093 

.£09 

.829 

23 

611.2!J5 

.786251 

8.179 

2.C48 

24 

682.704 

.834232 

.324 

.833 

24 

610.214 

.785482 

8.194 

2.052 

23 

681.454 

.S33t$;'3 

.338 

.837 

25 

609.136 

.784714 

8.208 

2.056 

26 

680.010 

.832515 

.353 

.840  1 

26 

608.062 

.783948 

8.223 

2.060 

27 

678.671 

.831(360 

.367 

1.844 

27 

606.992 

.788188 

8.237 

2.003 

28 

677.338 

.8';OH05 

.382 

1.848 

23 

605.926 

.782420 

8.252 

2.  066 

29 

676.008 

.KtfJ!!.53 

.396 

1.851 

29 

604.864 

.781657 

K.266 

2.070 

30 

674.686 

2.821)102 

.411 

1.855 

30 

603.805 

2.780897 

8.281 

2.074' 

31 

673.369 

2.828253 

.425 

1.858  i 

31 

602.750 

2.780137 

8.295 

2.077 

32 

G72.056 

.827405 

.440 

1.862  i 

32 

601.698 

.779379 

8.310 

2.081 

33 

WO.  748 

.836500 

.454 

.866  i 

33 

600.651 

.778622 

8.324 

2.084 

34 

600.440 

.8*5715 

.469 

.869 

34 

599.607 

.777867 

8.339 

2.088 

35 

668.148 

.834878 

.483 

.873 

35 

598.567 

.777112 

8.853 

2.092 

36 

666.856 

.824032 

.598 

.877 

36 

597.530 

.776360 

8.  £68 

2.096 

37 

665.668 

.823193 

.512 

.880 

37 

596.497 

.775608 

8.382 

2.099 

38 

004.  281  i 

.822855 

.527 

.884 

38 

595.467 

.774858 

8.397 

2.103 

39 

61  13.  008 

.821519 

.541 

.887 

39 

594.441 

.774109 

8.411 

2.106 

40 

661.736 

2.S2U685 

.556 

.892 

40 

593.419 

2.773361 

8.426 

2.  110 

41 

660.468 

2.819852 

.570 

.895 

41 

592.400 

2.772615 

8.440 

2.113 

42 

0.59.  :>(« 

.819021 

.585 

.899 

42 

591.384 

.771870 

8.455 

2.117 

43 

657.947 

.818191 

.599 

.903 

43 

590.37'2 

.771126 

8.469 

2.12j. 

.    44 

656.694 

.817363 

.614 

.906 

44 

589.364 

.770383 

H.484 

2.125 

45 

655.446 

.816537 

.628 

.910 

45 

588.359 

.769642 

8.498 

2.128 

46 

654.202 

.815712 

.643 

.914 

46 

587.357 

.768002 

8.513 

2.132 

47 

652.963 

.814889 

.657 

.918 

47 

586.359 

.768164 

8.527 

2.135 

48 

651.729 

.8140(37 

.<;7'2 

.921 

48 

585.364 

.767426 

8.542 

2.139 

49 

650.41)9 

.813247 

.686 

.924 

49 

584.373 

.766690 

8.556 

2.142 

50 

649.214 

2.812428 

.701 

.928 

50 

583.385 

2.765955 

8.571 

2.147 

51 

648.054 

2.811611 

.715 

.932 

51 

582.400 

2.765221 

8.585 

2.150 

52 

646.838 

810796 

.730 

.9£5 

52 

581.419 

.764489 

8.600 

2.154 

53 

645.627 

.7'44 

.9£9 

53 

580.441 

.763758 

8.614 

2.158 

54 

644.420 

.809169 

.759 

.943 

54 

579.466 

.763028 

8.629 

2.161 

55 

643.218 

.808358 

.773 

9-6  j 

55 

578.494 

.762299 

8.643 

2.105 

56 

642.021 

.807549 

.788 

.950 

56 

577.526 

.761572 

8.658 

2.168 

57 

640.828 

.806741 

.802 

.953 

57 

576.561 

.7'60845 

8.67'2 

2  172 

58 

639.639 

.805935 

.817 

.957 

58 

575.599 

.760120 

8.687 

2.175 

59 

638.455 

.805130 

.831 

.961 

59 

574.641 

.759397 

8.701 

2.179 

GO 

637.275 

2.804327 

.846 

.965 

60 

573.686 

2.758674 

8.716 

2.183 

284 


TABLE  IV.— RADII,  LOGARITHMS,  OFFSETS,  ETC. 


Deg. 

Radius. 

Loga- 
rithm. 

Tang. 
Off 

Mid. 
Ord. 

Deg. 

Radius. 

Loga- 
rithm. 

Tang. 
Off. 

Mid. 
Ord. 

I). 

R. 

log.  R. 

t. 

in. 

D. 

K. 

log.  K. 

t. 

in. 

10°  0' 

573.686    2.758674 

8.716 

2.183 

12°  0' 

478.339    2.679735 

10.453 

2.020 

2 

571.784 

.757232 

8.745 

2.190 

2 

477.018  j    .G7NW-> 

10.482 

2.028 

4 

569.896 

.755796 

8.774 

2.198 

4 

475.7t)5      .077338 

10.511 

2.0:35 

6 

508.020 

.754:304 

8.803 

2.205 

6 

47'4.400  !    .070145 

10.540 

2.042 

8 

566,166 

.752937 

8.831 

2.212 

8 

473.102      .074954 

H-.569 

2.050 

10 

504.305 

.751514 

8.800 

2.219 

10 

471.810 

.073707 

10.597 

2.657 

12 

502.406 

.750096 

8.889 

2.227 

12 

470.526      .072584 

10.020 

2.004 

14 

660.688 

.748683 

8.918 

2.234 

14 

409.249 

.671403 

10.055 

2.671 

16 

558.823 

.747274 

8.947 

2.241 

16 

467.978      .070226 

10.684 

2.079 

18 

557.019 

2.745870 

8.976 

2.234 

18 

466.715  12.609052 

10.713 

2.686 

20 

555.227 

2.744471 

9.005 

2.256 

20 

405.459    2.607881 

10.742 

2.093 

22 

553.447 

.743076 

9.034 

2.263 

22 

404.209 

.600713 

10.771 

2.701 

24 

551.678 

.741686 

9.003 

2.27'0 

?4 

402.900      .605549 

io.eoo 

2.708 

20 

549.920 

.740300 

9.092 

2.278 

26 

401.729  !    .004387 

10.829 

2.715 

28 

548.174 

.7:38918 

$.121 

2.285 

28 

400.500  I     (i(jf<22!) 

10.858 

2.722 

30 

546.438 

.737541 

9.150 

2.293 

30 

459.276  !    .602074 

10.887 

2.730 

32 

544.714 

.7'36169 

9.179 

2.300 

32 

458.000      .0001)22 

10.916 

2.737 

34 

543.001 

.734800 

9.208 

2.307 

34 

450.850      .<if>!»77:$ 

10.945 

2.744 

36 

541.298      .733436 

9.237 

2.314 

36 

455.040      .U5W*S 

10.973 

2  752 

38 

539.006    2.732077 

-9.266 

2.3*1 

38 

454.449  i2.  057485 

11.002 

2.759 

40     537.924  !  2.  730721 

9.295     2.329 

40 

453.259    2.050345 

11.031 

2.706 

42     526.253 

.729370 

9.324      2.330 

42 

452.073      .055208 

11.000 

2.774 

44     534.593 

.728023 

9.353 

2.343 

44 

450.894  i    .054075 

11.089 

46     532.943 

.7'26681 

9.382 

2.351 

46 

449.722      .052944 

11.118 

2  788 

48  1  531.303 

.725342 

9.411 

2.358 

48 

448.550 

.051810 

11.147 

2!  795 

50      529.673 

.724008 

9.440 

2.365 

EO 

447.395 

.050G91 

11.170 

2.803 

52     528.053 

.722677 

9.409 

2.372 

52 

446.241 

.64<J570 

11.205 

2.810 

54 

526.443 

.721:351 

9.498     2.  £-80 

.r4 

445.093 

.648451 

11.234 

2.S17 

56 

524.843 

.720029 

9.527 

2.387 

C6 

443.951 

.647  £35 

11.203 

2.825 

58 

523.252 

2  .718711 

9.556 

2.394 

58 

442.814 

2.040x21 

11.291 

2.832 

11°  o' 

521.671 

2.717397 

9.585 

2.402 

13°  0' 

441.  C84    2.045111 

11.820 

2.839 

2 

520.100 

.710087 

9.614      2.409 

2     440.559      .644004 

11.349 

2.840 

4 

518.539 

.714781 

9.642     2.416 

4 

439.440      .642899 

11.378 

2.854 

6 

516.986 

.713479 

9.671      2  423 

6     438.326      .641798 

11.407 

2.801 

8 

515.443 

.712181 

9.700     2.431 

8 

437.219      .04C099 

11.430 

2.868 

10 

513.909 

.710887 

9.729     2.438 

10 

430.117      .63CG03 

11.405 

2.870 

12 

512.385 

.  709596 

9  758     2.445 

12     435.020      .638510 

11.494 

2.883 

14 

510.869 

.708310 

9.787     2.453  i 

14 

433.929  !    .037419 

11.523 

2.890 

16 

509.363 

.707027 

9.816 

2.460  i 

16 

432.844      .636331 

11.552 

2.!:  98 

18 

507.865 

2.705748 

9.845 

2.467 

18 

431.T(54    2.0o524G 

11.  £80 

2.905 

20 

506.376 

2.704473 

9.874 

2.475 

20 

430  680  12.634104 

11.C09 

2.912 

22 

504.896 

.703202 

9.903 

2.482  i 

22 

429.020  1    .63:105 

11  .688 

2.919 

24 

503.425 

,701934 

9.932 

2.489 

24 

428.557        6321108 

11.C67 

2.127 

26 

501.902 

.700671 

9.961 

2.496 

26 

427.498      .6801I34 

ii.c.u; 

2.934 

28 

500.507 

.099410 

9.990     2.504 

28 

426.445 

.629868 

11  75.5 

2.941 

30 

499.001 

.098154    10.019     2.511 

30 

425.396 

.628794 

11.  7  £4 

2.949 

32     497.624 

.096901  1  10.  048 

2.518  ! 

32     424.354 

.627728 

11.783 

2  1>56 

34     496.195 

,695052  i  10.  077 

2.526  ! 

34     423.316 

.020005 

11.K12 

2.1JG3 

36 

494.774 

.694407 

10.106 

2.5S3 

36 

422.288 

025004 

11.  WO 

2.971 

38 

493.301 

2.693165 

10.135 

2.540 

38 

421.250 

2.024540 

11.809 

2.978 

40 

491.950 

2.691926 

10.164 

2.547 

40 

420.2-33 

2.023490 

11.898 

2.985 

42 

490.559 

.690692  ;10.192     2.555  ; 

42     419  215 

.622437 

11.927 

2.992 

44 

489.171 

.689460  J10.221      2.562  j 

44     418.203 

.621:387 

11.856 

3.000 

46 

487.790 

.688233    10.250 

2.509 

46     417.195 

.02033!) 

11.685 

3.007 

48 

480.417 

.687008    10  279 

2.577 

48     416.192 

.019294 

12.014 

3.014 

50 

485  051 

685788  110.308 

2.584 

CO     415.194 

.018251 

12.043 

3.022 

52 

483.094 

.684570  !  10.  337 

2.591 

52 

414.201 

.017211 

12.071 

3.029 

54 

482.344 

.683357  ;10.306 

2.598 

64 

413.212 

.010173 

12.100 

3.036 

56 

481.001 

.682146  110  395 

2.606 

56 

412.229 

.015138 

12.129 

3.044 

58 

479.000 

.680939  j  10.  424 

2.613  ! 

58  1  411.250 

.614106 

12.158 

3.051 

60  i  478.339   2.679735  !l0.453 

2.620  ! 

60  i  410.275  i  2.  613075 

12.187 

3.058 

285 


TABLE  IV.— RADII,  LOGARITHMS,  OFFSETS,  ETC. 


Deg. 
I). 

Radius. 

Loga- 
rithm. 

log.  R. 

Tan. 
Oil. 
t. 

Mid. 
Orel. 

111. 

Deg. 
B. 

Radius. 
K. 

Loga- 
rithm. 

log.  R. 

Tan. 

Off. 

t. 

Mid. 
Ord. 

m. 

14°  0' 

410.273  2.613075 

12.187 

3.058 

16°   ('•'    359.265 

2.555415 

13.917     3.496 

2 

409.  8  Jtf     .612048 

12.216 

3.065 

2     358.528 

.554517 

13.946     3.504 

4 

408.341 

.611023 

12.245 

3.073 

4     a57.784 

.553621 

13.975     3.511 

6 

407.390 

.610000 

12.274 

3.080 

6     357.048 

.552727 

14.004     3.518 

8 

408.424     .603980 

12.302 

3.087 

8    a56.31i; 

.551834 

14.0a3    3.526 

10 

40.1.473     .607962 

12.331 

3.095 

10  [355.585 

.550944 

14.061     3.533 

12 

404.528      .606946 

12.360 

3.102 

12 

354.859 

.550055 

14.090     3.540 

14 

403.583 

.605933 

12.389    3.109 

14 

354.1:35 

.549169 

14.119     3.547 

10 

403.645 

.604923 

12.418    3.117 

16 

a53.414 

.548284 

14.148 

3.555 

18 

401.712 

.603914 

12.447 

3.124 

18 

352.696 

.547401 

14.177 

3.562 

20 

400.782  2.602903 

12.476 

3.131 

20    351.981 

2.546519 

14.205 

3.569 

22 

899.857 

.601905 

12.504    3.1:38 

22     351.269 

.545640 

14.234 

3  .  577 

2i 

398.937 

.603904     12.  533  j  3.  146 

24 

350.560 

.544762 

14.263 

3.584 

26 

398.020 

.59J905     12.562    3.153 

26 

349.854 

.543887 

14.292 

3.591 

28 

397.108 

.59893$     12.591    3.160 

28    349.150 

.543013 

14.320 

3.599 

30 

398.200 

.597914     12.620    3.168 

30    348.450 

.542140 

14.349 

3.606 

32 

39.-).  29ti 

.59692:2     12.649 

3.175 

32 

347.  75£ 

.541270 

14.378 

3.613 

34 

391.396 

.595933     12.678 

3.182 

34 

347.057 

.540401 

14.407 

3.621 

36 

393.501 

.594945     12.  708  13.190 

36 

346.365 

.539535 

14.436 

3.628 

38 

392.609 

.593960     12.  735  j  3.197 

38 

345.676 

.538670 

14.464 

3.635 

40 

391.722  2.  592:)  78     12.764    3.204 

40    344.990 

2.537806 

14.493 

3.643 

42 

390.83$'   .5919'.)7     12.793:3.211 

42     344.306 

.  53(5945 

14.522 

3.650 

44 

389.959 

.591019  ,  12.822 

3.219 

44     343.625 

.536085 

14.551 

3.657 

46 

389.084 

.590043     12.851 

3.2215 

46 

342.947 

.5:35227 

14.580 

3.664 

48 

388.212 

.5890J9     12.830    3.233 

48 

342.271 

.  534370 

14.608 

3.672 

50 

837.845 

.5330.)  7     12.908 

3.241 

50     341.598 

.533516 

14.637 

3.679 

52 

888.481 

.587128     12.937 

3.243 

52 

340.  92  i 

.5:32663 

14.666 

3.686 

54 

885.621 

.586161     12.986 

3.255 

54     340.260 

.531811 

14.695 

3.694 

56 

384.765 

.585196     12.99") 

3.263 

58     339.595 

.530962 

14.723 

3.701 

5i 

383.913 

.584233     13.024 

3.270 

58    338.933 

.530114 

14.752 

3.708 

15a   0 

833.035 

2.583272     13.053 

3.277 

17°   0 

338.273 

2.529268 

14.781 

3.716 

2 

83*.  2*0 

.532314     13.031 

3.284 

2 

337.616 

.528424 

14.810 

3.723 

4 

881.  88  J 

.581358     13.110 

3.21)2 

4 

3:36.982 

.527581 

14.838 

3.7:30 

6 

380.543 

.580403     13.139 

3.299 

6 

336.310 

.526740 

14.867 

3.738 

8 

37;).  70.) 

.579451     13.163 

3.30J 

8 

335.6601   .525900 

14.896 

3.745 

10 

378.8*1 

.578501     13.197 

3314 

10 

335.013 

.525062 

14.925 

3.752 

12 

378.051 

.577553     13.22) 

3.3-21 

12 

334.369 

.524226 

14.954 

3.760 

14 

377.251 

.576608     13.254 

3.323 

14 

333.727 

.523392 

14.982 

3.767 

1(5 

376.412 

.575664     13.233 

3.336 

16 

333.033 

.522,559 

15.011 

3.774 

18 

375.597 

.574722     13.312 

3.343 

18 

332.451 

.521728 

15.040 

3.781 

20 

374.788 

2.573783     13.341 

3.350 

20 

331.816 

2.520898 

15.069 

3.789 

22 

373.977 

.572845     13.370    3.353 

22 

331.184 

.520070 

15.097 

3.796 

24 

373.173 

.571910     13.399    3.365 

24 

330.555 

.519244 

15.126 

3.803 

26 

872.372 

.570977     13.427 

3.372 

26 

329.923 

.518419 

15.155 

3.811 

28 

371.574 

.570045     13.455 

3.  >79 

28 

329.303!   .517596 

15.184 

3.818 

30 

370.780 

.569116     13.435 

3.337 

30 

323  639 

.516774 

15.212 

3.825 

32 

369.  93  ') 

.568189     13.514 

3.394 

32     323.061 

.515954  15.241 

3.8:33 

31 
36 

361).  202 
368.418 

.567264     13.543 
.566340     13.572 

3.401 
3.409 

34     327.413 

36     32li.  8-28 

515138;  15.270 
.514:319|  15.299 

3.840 
3.847 

38 

367.  63  7 

.565419     13.60) 

3.416 

38 

326.215 

.'5ia504  15.327 

3.855 

40 

868.859 

2.564530     13.629 

3.423 

40 

325.604 

2.512690  15.a56 

3.862 

42 

366.085 

.563532 

13.658 

3.431 

42 

324.996 

.511878!  15.385 

3.869 

44 

365.315 

JS63367 

13.687 

3.438 

44 

324.390 

.511067  15.414 

3.877 

46 

364.547 

.561754 

13.716 

3.445 

46 

323.736 

.510258  15.442 

3.884 

48 

363.783 

.560343 

13.744 

3.452 

48 

323.184 

.509451 

15.471 

3.891 

50 

363.022 

.559933 

13.773 

3.460 

50 

322.585 

.508645 

15.500 

3.899 

52 

362.264 

.559028 

13.802 

3.467 

52 

321.939 

.507'840  15.528 

3.906 

54 

361.510 

.558120 

13.831 

3.474 

54     321.394 

.507037  15.557 

3.013 

56 

360.758 

.557216 

13.860 

3.482 

56 

320.801 

.50(5236  15.586 

3.920 

58 

360.010 

.556315 

13.889 

3.489 

58 

320.211 

.505436 

15.615 

3.928 

60 

359.265 

2.555415 

13.917    3.496'         60  1319.623 

2.504638  15.643 

3.935 

280 


TABLE  IV.— RADII,  LOGARITHMS,  OFFSETS,  ETC. 


Deg. 

Radius. 

Loga- 
rithm. 

Tang. 

Off. 

Mid. 
Ord. 

Deg. 

«•*"-  ias: 

Tang. 
Off. 

Mid. 
Ord. 

IX 

R. 

log.R. 

t. 

m.   i 

IX 

R.       log.  R. 

t. 

m. 

18°  0' 

319.623  2.504638 

15.643    3.985     20°  0' 

287.  939  !  2.  459300 

17.365 

4.374 

2 

319.037     .503841 

15.672 

3.942   1          10 

i  285.583 

.4557:33 

17.508 

4  411 

4 

318.453 

.503045 

15.701 

3.950 

20 

283.267 

.452195 

17.651 

4.448 

6 

317.871 

.502251 

15.730 

3.957 

30 

280.988 

.448688 

17.794 

4.484 

8 

317.292 

.501459 

15.758 

3.964 

40 

278.746 

.445209 

17.937 

4.521 

10 

316.715 

.500668 

15.787 

3.972 

50 

1  276.541 

.441759 

18.081 

4.55S 

12 

316.139 

.499879 

15.816    3  979     21°  ()' 

274.370 

2.438337 

18.224 

4.594 

14 

315.566 

.499091 

15.845    3.986  N          10 

272.234 

.434943 

18.367 

4.631 

16 

314.993     .498:304 

15.873    3.994 

20 

271.032 

.431576 

IS.  509 

4.668 

18 

314.42ii 

.497519 

15.902 

4.001! 

30 

268.002 

18.652 

4.704 

20 

22 

313.860  2.496736 

313.295!    .495953 

15.931    4.008! 
15.959    4  016  i 

40 
50 

266.024     .424921 
264.018     .421633 

18.795 
18.938 

4.741 
4.778 

24 

312.732 

.495173 

15.988    4.023     Q2°  0' 

262.042 

2.418371 

19.081 

4.814 

26 

312.172 

.494393 

16,017 

4.030            10 

260.098     .4151:34 

19.224 

4.851 

28 

311.613 

.493616 

16.046 

4.038            20 

258.180 

.411922 

4  888 

30 

311.056 

.492839 

16.074 

4.045'           30 

i  256.292 

408734 

19!  509 

4.925 

32 

310.502 

.492064 

16.103 

4.052i;         40 

254.431      .405571 

19.052 

4.961 

34 

309.949 

.491291 

16.132    4.060  j         50 

252  599  !     402431 

19.7SI4 

4.998 

36 

309.399 

.490518 

16.160 

4.  067!!  23°  0' 

250.  793  '•  2.  399:315 

19.937 

5.035 

38 

308.850     .489748 

16.189 

4.074 

10 

249.013!    .39Q222 

20.079 

5.  Oil 

40 

308  303  !2  488978 

16.218    4  081 

247.258     .393151 

20.222 

5.108 

42 
44 

307.759 
307.216 

.488210 
.487444 

16.246    4.089i| 
16.275    4.09(5 

1245.473;   .390103 
243.825     .387077 

20.364 
20.507 

5.145 
5.182 

46 

306.675 

.486679 

16.304 

4.103 

242.144 

.384074 

20.64J 

5.218 

48 
50 
52 
54 
56 
58 

30(5.136 
305.599 
305.064 
304.531 
304.000 
303.470 

.485915 
.485152 
.484391 
.483632 
.482873 
.482116 

16.333 
16.361 
16.390 
16.419 
16.447 
16.476 

4.111 
4.118 
4.125 
4.133 
4.140 
4.147 

24°  0';  240.  487 
10:238.853 
20    237.241 
30    235.052 
40    234.084 
50    232.537 

2.381091 
.378130 
.375190 
.372270 
.369371 
.3(56492 

20.791 
20.933 
21.076 
21.218 
21.360 
21.502 

5.255 
5.292 
5.329 
5.306 
5.402 
5.439 

19°  0' 

302.943  2.481361 

16.505 

4.155 

25°  0' 

231.011 

2.363633 

21.644 

5.476 

2 

302.417 

.480607 

16.533 

4.162 

10 

229.50(3 

.360794 

21.786 

5.513 

4 

301.893 

.479854 

16.562 

4.169 

20 

228.020 

.357974 

21.928 

5.549 

-6 

301.371 

.479102 

16.591    4.177! 

30 

'226.555 

.355173 

22.070 

5.586 

8 

'300.851 

.478:352 

16.  620  !  4.  184! 

40 

225.108 

.352391 

22.212 

5.623 

10 

300.333 

.477(503 

16.  648  14.  191  j 

50 

223.680 

.349627 

22.353 

5.660 

12 
14 
16 

18 

i  299.  816 
299.302 

298.789 
!  298.278 

.476855 
.476109 
.475364 
.474621 

16.  677  j  4.  199' 
16.  706!  4.  206 
16.734    4.213 
16.  763|  4.  221 

26°  0 

10 
20 
30 

222.271 

220.879 
219.506 
218.150 

2.346882 
.344155 
.341446 
.338755 

22.495 
22.1537 
22.778 
22.920 

5.697 
5.734 
5.770 
5.807 

20 

1297.768 

2.473878 

16.792    4.228 

40 

216.811     .336081 

23.062 

5.844 

22 

1297.260 

.473137 

16.8201  4.  235 

50 

'215.489      .333424 

23.203 

5.881 

24 

'296.755 

.472398 

16.849 

4.243l;27°  0 

:  214.  183  2.330785 

23.345 

5.918 

26 

296.250 

.471659 

16.878 

4.250   !         10 

212.893;    .328162 

23.  4S6 

5.955 

28 

295.748 

.470922 

16.906 

4.257 

20 

211.620     .325556 

23.627 

5.992 

30 

i  295.247 

.470186 

16.!  135    4.265 

30 

210.362     .322967 

23.7(59 

6.  (129 

32 

1294.748 

.469452 

16.964   4.272            40 

209.119 

.320:393  23.910 

6.065 

34 

294.251 

.468718 

1   .992   4.279  i         50 

207.891 

.317836 

24.051 

6.102 

36 

38 

29:3.756     .467986 
293.262     .467256 

'oS  '  1'Sl     28°  0'  206.678  12.315295  24.192 
10    205  .  480     .  312769  24  .  333 

6.139 
6.176 

40 

292.770  2.466526 

.078  !  4.301            20  !  204.296     .310259  24  .47'4 

6.213 

42 

292.279     .4(55798 

.10714.308            30 

1203.125     .307764 

24.615 

6.250 

44 

291.790     .465071 

.186  4.816           40 

1  201.  969;    .305285 

24.756 

6.287 

46 

291.303 

.464345 

.164    4.323            50 

200.826  :    .302820 

24.897 

6.324 

48 

290.818 

.463621         ,198  [4.880     29°  0'  199.696  2.300370  25.038 

6.360 

50 

290.  334 

.462897     1   .222    4.338            10 

198.580 

.297935 

25.179 

6.398 

52 

289.  851 

.462175 

.250    4.345            20 

197.476 

.295515  25.320 

6.435 

54 

!  289.371 

.461455         .279    4.352            30 

196.  3&5 

.293108 

25.4CO 

6.472 

56 

:  288.892 

.460735         .308    4.360  1         40 

195.306 

.290716  25.601 

6.509 

58 

i  288.414 

.460017 

17.336    4.367;i         50 

194.240     .288338 

25.741 

6.545 

60 

'<  287.939  2.459300 

17.365    4.374     80°  0 

193.185!  2.  285974 

25.882 

6.583 

287 


TABLE  IV.— RADII,  LOGARITHMS,  OFFSETS,  ETC. 


! 

1 

Deg.    Radius. 

Loga-     Tang.  '  Mid. 
rithm.       Off.      Ord. 

Deg.    Radius. 

Loga- 
rithm. 

Tang. 
Off? 

Mid.     I 
Ord.     1 

D. 

B. 

log.  R. 

t. 

in. 

D. 

K. 

log. 

K. 

t. 

in. 

30°  20'   191.111 

2.281286    5 

J6.163 

6.657 

38^ 

30' 

151.657 

2.180863 

32 

.969 

8.479 

4 

0    18 

J.083 

.2701 

552    ; 

26.443    6.731 

V 

• 

49.787 

.175 

t::. 

33 

.381 

8.592 

31° 

V    187.099 

.272071     26.724 

6.805 

30 

147.965 

.170160 

88 

.792 

8.704 

X 

t)     18 

5.158 

.267.' 

>41  15 

27.004 

ft.  879 

40< 

()' 

1 

46.190 

.164 

918 

84 

.202     8.816 

--  A 

0     18. 

1.258 

.26* 

)62     j 

27.284 

6.958 

30 

1 

44.460 

.15£ 

747 

34 

.612     8.929 

32° 

0'    181.398 

.258632     27.564 

.027 

41 

«  0' 

142.773 

.154645 

35.024 

9.041 

2 

0     17 

J.577 

.254; 

250     , 

27.843 

.101 

30 

1 

41.127 

.14S 

6i< 

35 

.429 

9.154 

40     17 

7.794 

.249916     { 

28.123 

.175 

42°  0'  i 

39.521 

.144641 

35.837 

9.267 

33° 

(V      7 

E5.047 

.245( 

528     , 

28.402 

.250 

:>'.' 

1 

37.97)5 

i  •>( 

36 

.244 

9.380 

t 

0       7 

L336 

.241: 

58(5 

28.680 

.324  ;  43 

5  0' 

1 

3(5.425 

.134 

syr 

:,(i 

.650 

9.493 

40       72.659 

.237188  i  28.959 

.398  : 

80 

134.932 

.130114  37 

.056 

9.606 

34° 

(V      7 

1.015 

2.23:1035     29.237 

.473 

44 

'  (i 

133.4732.125395  37 

.461 

9.719 

2 

0     f6 

J.404 

.228( 

)24 

29.515 

.547  i 

30 

1 

32.049     .12( 

1784 

;;; 

.865 

9.8512 

4 

0     16 

7.825 

.224S 

65 

29.793 

.621 

45 

=  0 

1 

30.6561    .IK 

i:;i 

38 

.268 

9.946 

35° 

0'    1(5 

3.  275 

.220 

08 

10.071 

30 

1 

29.296!    .111 

584 

88 

.671 

10.059 

20     1(54.75(5 

.216842 

10.348    7.770 

46 

»  0'   127.9651    .KV7092  39.073 

10.173 

4 

0     16. 

1.2(50 

.212 

W5 

10.625 

7.845 

20 

1 

26.664     .10$ 

!65£ 

81 

.474 

10.286 

36° 

0'    161.803 

.20S988     30.902 

7.919  !  47 

125.392 

.098270  3! 

.875 

10.400 

2 

o   ie 

0.3(58 

.205 

119 

11.178 

7.994 

80 

1 

24.148     .09f 

!'.i:> 

-l< 

.275 

10.516 

4 

0     15 

8.960 

.201 

388 

11.454 

8  068  1  ,  48 

°     0 

1 

22.930!    .08' 

»•;.->; 

41 

.674 

10.628 

37° 

0'    15 

7.577 

.lit;  494     31.730 

80 

121.738!    .085425  41 

.072 

10.742 

20     156.220 

.193736     32  006 

8*218     49°  0'   120.5711    .  0812-13  41<469 

10.856 

40  :  154.8S7 
38°   0';  153.578 

.190014  i  32.  282 
2.186328     32.557 

8.292 
8  36; 

50 

SO    119.429     .077109  41.866 
0  0'   118.3102.073022  42.262 

10.970 
11.085 

I 

TABLE  V.  -CORRECTIONS  FOR  TANGENTS  AND  EXTERNALS. 

FOR  TANGENTS,  ADD 

FOR  EXTERNALS,  ADD 

Ang 

5° 

10° 

15° 

20° 

25° 

30° 

Ang 

5° 

10° 

15° 

2 

0° 

25° 

30° 

A 

Cur. 

Cur.  Cur. 

Cur. 

Cur. 

Cur. 

A 

Cur. 

Cur.  Cur. 

Cur. 

Cur 

.    Cur. 

10° 

.03 

.06       .09 

.13 

.16      .19 

10° 

001 

.003 

.004 

.006 

.007     .008 

20 

.06 

.13 

.19 

.26 

.32 

.39 

20 

.OC 

Mi 

.011 

.017 

.( 

23 

.028     .034 

30 

.10 

.19 

.29 

.39 

.49 

.59 

30 

.0 

8 

.025 

.038 

.( 

51 

.065     .078 

40 

.13 

.26 

.40 

.53 

.67 

80 

40 

M 

;:i 

.046 

.07'0 

.C 

98 

.117     .141 

50 

.17 

.34 

.51 

.68 

.85    1  02 

50 

.()• 

g 

.075 

.116 

.1 

61 

.189     .227 

60 

.21 

.42 

.63 

.84 

1.051  1.27 

60 

.0. 

,i; 

.112 

.168 

1   .S 

25 

.28.' 

J     .340 

70 

.25 

.51 

.76 

1.02 

il.28jl.54 

70 

.0 

«» 

.159 

.240 

.321 

.40; 

J     .485 

80 

.30 

.61 

.91 

1.22 

1.53    1.84  i  80 

.1 

0 

.220      .332 

1  -4 

45 

.558  !   .671 

90 

.36 

.72 

1.09 

1.45 

1.83    2.20      90 

.149 

.299      .450 

M 

.0:1 

.756     .910 

100 

.43 

.86 

1.30 

1  74 

2  18  I  2.62 

100 

.2( 

(0 

.401 

.604 

1  .1 

'(19 

1.015   1.221 

110 

.51 

1.Q8 

1.56 

2  08 

2.61    3.14 

110 

.8 

is 

.536      .806 

i.( 

>S2 

1.S55   1.633 

120 

.62 

1.25 

1.93 

2.52 

3  16 

3.81 

120 

.3 

50 

.721  1.086 

1.456 

1.82. 

5  2.197 

] 
i 

288 


TABLE  VI.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


r" 
j 

II 

Angle. 

Tan- 
gent. 

Exter- 
nal. 

Angle. 

Tan- 
gent. 

Exter- 
nal. 

Angle. 

Tan- 
gent. 

Exter- 
nal. 

A 

T. 

E. 

A 

T. 

E. 

A 

T. 

E. 

1° 

50.00 

.218 

11 

551.70 

26.500 

81° 

1061.9  i     97.577 

10' 

58.  iW 

.297 

10'  1.    560.11 

27.313 

10'    1070.6       99.155 

20 

60.07 

.388 

20       568.53      28.137 

20      1079.2  :  100.75 

30 

75.01 

.491 

30  '     576.95      28.974 

30  i  1087.8      102.35 

40^ 

83.34 

.606 

40       585.36 

29.S24 

40 

1096.4      103.97 

50 

91.68 

.733 

50 

593.79 

30.686 

50 

1105.1      105.60 

2 

100.01 

.873 

12 

602.21 

31.561 

22 

1113.7      107.24 

10 

108.35 

1.024 

10 

610.64      32.447 

10 

1122.4      108.90 

20 

116.68 

1.188 

20       619.07 

as.  347 

20 

1131.0      110.57 

30  !  125.02 

1.364 

30       627.50 

34.259 

30 

1139.7  :  112.25 

40      133.36 

1.552              40 

6:35.93      35.183 

40 

1148.4  :  113.95 

50 

141.70 

1.752 

50 

644.37     36.120 

50 

1157.0  !  115.66 

3 

150.04 

1.964 

13             652.81  i  37.070      23 

1165.7     117.38 

10 

158.38 

2.188 

10       661.25  ;  38.031  i           10 

1174.4  i  119.12 

20 

166.72 

2.425  !           20       669.70      39.000              20 

1183.1  1  120.87 

30 

175.06 

2.674  !           30       678.15  :  39.993  .            30 

1191.8  !  122.63 

40 

183.40 

2.934  j           40       686.  6.»      40.993  !!          40 

1200.5      124.41 

50 

191.74 

3.207  i           50       695.06  !  42.004  ;!          50 

1209.2  i  126.20 

4 

200.08 

3.492    'l4 

703.51  <  43.029      24 

1217.9      128.00 

10 

208.43 

3.790    i          10 

711.97     44.066              10 

1226.6  1  129.82 

20 

216.77 

4.099              20       720.44     45.116              20 

1235.3      131.65 

30 

225.12 

4.421 

30       728.90     46.178             30 

1244.0      133.50 

40 

233.47 

4.755 

40 

737.37     47.253             40 

1252.8      135.  ,35 

50 

241.81 

5.100 

50 

745.85     48.341  ;|          50 

1261.5  !  137.23 

5 

250.16 

5.459 

15 

754.32     49.441      25 

1270.2  j  139.11 

10 

258.51 

5.829 

10 

762.80     50.554              10 

1279.0      141.01 

20 

266.86 

6.211 

20 

771.99     51.679             20 

1287.7      142.93 

30 

275.21 

6.606 

30 

779.77     52.818 

30 

1296.5      144.  85 

40 

283.57 

7.013 

40 

788.26 

53.969 

40 

1305.3 

146.79 

50 

291.92 

7.432 

50 

796.75 

55.132 

50 

1314.0      148.75 

6 

300.28 

7.863 

16 

805.25     56.309 

26 

1322.8      150.71 

10 

308.64 

8.307 

10 

813.75     57.498 

10 

1331.6      152.69 

20 

316.99 

8.762 

20 

822.25 

58.699 

20 

1340.4      154.69 

30 

325.35 

9.230 

30 

830.76 

59.914 

30 

1349.2      156.70 

40 

333.71 

9.710 

40 

839.27     61.141 

40 

1358.0 

158.72 

50 

342.08 

10.202 

50 

847.78     62.381             50 

1366.8 

160.76 

7 

&50.44 

10.707 

17 

856.30     63.634 

27 

1375.6     162.81 

10 

&58.81 

11.224 

10 

864.82     64.9(30 

10 

1384.4      104.86 

20 

367.17 

11.753 

20 

873.35 

66.178    :          20 

131)3.3      166.95 

30 

375.54 

12.294 

30 

881.88     67.470              30      1402.0      1189.04 

40 

383.91 

12.847 

40 

890.41      68.774  :           40      1410.9      171.15 

50 

392.28  :  13.413 

50 

898.95  ;  70.091  i,          50 

1419.7 

173.27 

8 

400.66  i  13.991 

18 

907.49  i  71.421      28 

1428.6 

175.41 

10 

409.03 

14.582 

10 

916.03     72.764              10 

1437.4 

177.55 

20 

417.41 

15.184 

20 

924.58     74.119  !|          20 

1446.3 

179.72 

30 

425.79 

15.799 

30 

933.13     75.488    !          30 

1455.1 

181.89 

40 

434.17 

16.426 

40 

941.69      76.869              40 

1464.0 

184.08 

50 

442.55 

17.065 

50 

950.25     78.264             50 

1472.9 

186.29 

9 

450.93 

17.717 

19 

958.81     79.671    !  29 

1481.8 

188.51 

10 

459.32 

18.381 

10 

967.88     81.092 

10 

1490.7 

190.74 

20 

467.71 

19.058 

20 

975.96      82.525 

20 

1499.6 

192.99 

30 

476.10 

19.746 

30 

984.53     88.973  i 

30 

1508.5 

195.25 

40 

484.49 

20.447 

40 

993.12 

85.431 

40 

1517.4 

197.53 

50 

492.88 

21.161 

50 

1001.7 

86.904  i 

•50 

1526.3 

199.82 

10 

501.28 

21.887 

20 

1010.3 

88.389    !  30 

1535.3 

202.12 

10 

509.68 

22.624 

10 

1018.9 

89.888 

10 

1544.2 

204.44 

20 

518.08 

23.375 

20 

1027.5 

91.399 

20 

1553.1 

206.77 

30 

526.48 

21.138 

30 

1036.1 

92.924  ! 

30 

1562.1 

209.12 

40 

534.89 

24.913 

40 

1044.7       94.462 

40 

1571.0 

211.48 

50 
k. 

543.29 

25.700 

50 

1053.3    i  96.013 

50 

1580.0 

213.86 

TABLE  VI.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 

A 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

1 

Angle. 

A 

Tan- 
gent. 
T. 

Exter- 
nal. 

E. 

Anglo. 

A 

Tan- 
gent. 

T. 

Exter- 
nal. 
E. 

31° 

1589.0 

216.25  I 

41° 

2142.2     387.38 

51° 

2732.9 

618.39 

10 

1598.0 

218.66 

10' 

2151.7     390.71 

10' 

2743.1 

622.81 

20 

1606.9     221.08 

20 

2161.2     394.06 

20 

2753.4 

627.24 

30 

1615.9     223.51 

30 

2170.8 

397.43 

30 

2763.7 

631.69 

40 

1624.9     225.96 

40 

2180.3 

400.82 

40 

2773.9 

636.17 

50 

1633.9     228.42 

50     2189.9 

404.22 

50 

2784.2 

640.66 

32 

1643.0     2:30.90 

42 

2199.4 

407.64 

52 

2794.5 

645.17 

10 

1652.0  !  233.39 

10 

2209.0 

411.07 

10 

2804.9 

649.70 

20 

1661.0     2&5.90 

*    20 

2218.6 

414.52 

20 

2815.2 

654.25 

30 

1670.0 

2:38.43 

30 

2228.1 

417.99 

30 

2825.6 

658.83 

40 

1679.1 

240.96 

40 

2237.7 

421.48 

40 

2835.9 

663.42 

50 

1688.1  I  243.52 

50 

2247.3 

424.98 

50 

2846.3 

668.03 

33 

1697.2     246.08 

43 

2257.0 

428.50 

53 

28S6.7 

672.66- 

10 

1706.3     248.66 

10 

2266.6 

432.04 

10 

2867.1 

677.32 

20 

1715.3 

251.26 

20 

2276.2 

435.59  ! 

20 

2877.5 

681.99 

30 

1724.4 

253.87  ; 

30 

2285.9 

439.16  ;           30 

2888.0 

686.68 

40 

1733.5 

256.50 

40 

2295.6 

422.75  ! 

40 

2898.4 

691  .40 

50 

1742.6 

259.14 

50 

2305.2 

446.  &5  ; 

50 

2908.9 

696.13 

34 

1751.7 

261.80 

44 

2314.9 

449.98  : 

54 

2919.4 

700.89 

10 

1760.8 

264.47 

10 

2324.6 

453.62  i 

10 

2929.9 

705.66 

20 

1770.0 

267.16 

20 

2^334.3 

457.27  i 

20 

2940.4 

710.46 

30 

1779.1 

269.86 

30 

2:344.1      460.95 

30 

2951.0 

715.28 

40 

1788.2 

272.58 

40 

2353.8     464.64 

40 

2961.5 

720.11 

50 

1797.4 

275.31 

50 

2363.5     468.35 

50 

2972.1 

724.97 

35 

1806.6 

278.05 

45 

2373.3  !  472.08 

55 

2982.7 

729.85 

10 

1815.7 

280.82 

10 

2383.1 

475.82 

10 

2993.3 

734.76 

20 

1824.9 

283.60 

20 

2392.8 

479.59 

20 

3003.9 

739.68 

30      1834.1      280.39 

30 

2402.6 

483.37 

30 

3014.5 

744.62 

40      1843.3     289.20 

40 

2412.4 

487.17 

40 

3025.2 

749.59 

50      1852.5     292.02 

50 

2422.3 

490.98 

50 

3035.8 

754.57 

36            1861.7     294.86 

46 

2432.1  j  494.82 

56 

3046.5 

759.58 

10      1870.9     297.72 

10 

2441.9     498.67  i 

10 

3057.2       764.61 

20 

1880.1      300.59 

20 

2451.8  i  502.54 

20 

3067.9 

769.66 

30 

1889.4  !  303.47 

30 

2461.7     506.42 

30 

3078.7 

774.73 

40      1898.6     306.37 

40 

2471.5     510.33 

40 

3089.4 

779.83" 

50 

1907.9 

309.29 

50 

2481.4 

514.25 

50 

3100.2 

784.94 

37 

1917.1 

312.22 

47 

2491.3 

518.20 

57 

3110.9 

790.08 

10 

1926.4 

315.17 

10 

2501.2 

522.16 

10 

3121.7 

795.24 

20      1935.7 

318.13 

20 

2511.2 

526.13 

20 

3132.6 

800.42 

30  i  1945.0     321.11 

30 

2521.1 

530.13  ! 

30  !  3143.4       805.62 

40 

1954.3 

324.11 

40 

2531.1      534.15 

40  I  3154.2  :     810.85 

50 

1963.6 

327.12 

50 

2541.0     538.18 

50  1  3165.1        816.10 

38 

1972.9 

330.15 

48 

2.551.0 

542.23 

58         i  3176.0       821.37 

10 

1982.2  1  333.19 

10 

2561.0 

546.30  > 

10  i  Ml  86.  9 

826.66 

20     1991.5      336.25 

20 

2571.0  ]  550.39 

20     3197.8 

K3i  UK 

30     200J3.9      339.32 

30 

2581.0  |  554.50 

30     3208.8       837.31 

40     2010.2  1  342.41 

40 

2591.1  i  .'58.63 

40     3219.7 

842.67 

50 

2019.6 

345.52 

50 

2601.1      562.77 

.50     3230.7 

848.06 

39 

2029.0 

S48.64 

49 

2611.2     566.94 

59           3241.7 

853.46 

10 

20:38.4 

&51.78 

10 

2621.2     571.12 

10 

3252.7 

858.89 

20 

2047.8 

354.94 

20 

2631.3  1  575.32  i 

20 

3263.7  I     864.34 

30 

2057.2 

358.11 

30 

2641.4 

579.54  i 

30 

3274.8 

869.82 

40 

2066.6 

361.29 

40 

2651.5 

583.78  ' 

40  !  3285.8 

875.32 

50 

2076.0 

364.50 

50 

2661.6 

588.04  i 

50     3296.9 

880.84 

40 

2085.4 

367.72 

50 

2671.8 

592.32 

60           3308.0 

886.38 

10 

2094.9  I  370.95 

10 

2681.9 

596.62 

10     3319.1 

891.95 

20 

2104.3 

374.20  ; 

20 

2692.1 

600.93 

20  j  3330.3 

897.  B4 

30 

2113-.8 

377.47  | 

30 

2702.3 

605.27 

30 

3341.4 

903.15 

40 

2123.3 

380.76 

40 

2712.5 

609.62  i 

40 

3352.6 

908.79 

50 

2132.7  -  384.06  j 

50 

27,22.7 

614.00 

50 

3363.8 

914.45 

290 


TABLE  VI.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 

A 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

Angle. 

A 

Tan- 
gent. 

T. 

Exter- 
nal. 

«.  i 

i 
Angle. 

A 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

61° 

3375.0 

920.14       71° 

4086.9 

1308.2 

81° 

4893.6 

1805.3    ; 

10' 

3386.3 

925.85  ! 

10' 

4099.5 

1315.6 

10' 

4908.0 

1814.7 

20 

3397.5 

931.58 

20 

4112.1 

1322.9 

20 

4922.5 

1824.1 

30 

3408.8 

937.34 

30 

4124.8 

1330.3 

30 

4!;c7  0 

1633.6 

40 

3420.1        943.12 

40 

4137.4 

1337.7 

40 

4951  !  5 

1843.1 

8481.4 

948.92 

50 

4150.1 

1345.1 

50 

4966.1 

It  52  .  6 

62 

3442.7 

954.75       72 

4162.8 

1352.6 

82 

4980.7 

1E6&8 

10     3454.1 

960.60              10 

4175.6      1360,1 

10 

4995.4      1871.8 

20  i  3405.4 

966.48              20 

4188.5  j  1367.6 

20 

5010.0 

iesi.5 

30  !  3476.8 

972.38             30 

4201.2  I  1375.2 

30 

£024.8 

1F91.2 

40  :  3488.3 

978.31     !          40 

4214.0      1382.8 

40 

.5039.5 

18C0.9 

50 

13499.7 

984.27             50 

4226.8 

1390.4 

50 

5054.3 

1910.7 

*63 

3511.1 

990.24 

73 

4239.7 

1398.0  i!  83 

5069.2 

1920.5 

10 

3522.  G 

996.24 

10 

4252.6      1405.7              10 

£084.0  1  1930.4 

20 

3534.1 

1002.3                20 

4265.6      1413.5  i            20 

£099.0      1940.3 

30 

3545.6 

1008.3                30 

4278.5 

1421.2  j            30 

5113.9  i  1950.3 

40 

3557.2 

1014.4 

40 

4291.5 

1429.0  i            40 

5128.9  !  1SC0.2 

50 

3568.7 

1020.5                50 

4304.6 

1436.8              50 

5143.9  ;  1970.3 

64 

3580.3 

1026.6 

74 

4317.6 

1444.6  j    84 

5159.0  j  1980.4 

10 

3591.9 

1032.8 

10 

4330.7 

1452.5              10 

"5174.1      1990.5 

20 

3603.5 

1039.0 

20 

4343.8 

1460.4 

20 

5189.3     2C00.6 

30 

3615.1 

1045.2 

30 

4356.9 

1468.4 

20 

5204.4     2010.8 

40 

3626.8 

1051.4 

40 

4370.1 

1476.4 

40 

5219.7  i  2021.1 

50 

8638.5 

1057.7 

50 

4383.3 

1484.4 

50 

5234.9 

2031.4 

65 

3650.2 

1063.9 

75 

4396.5 

1492.4 

85 

5250.3 

2041.7 

10 

3661.9 

1070.2 

10 

4409.8 

1500.5 

10 

C-265.6  i  2052.1 

20 

3673.7 

1076.6 

20 

4423.1 

1508.6 

20 

5281.0  i  2062.5 

30 

3685.4 

1082.9 

30 

4436.4 

1516.7 

30 

££96.  4     2073.0 

40 

3697.2 

1089.3 

40 

4449.7 

1524.9 

40 

5311.9      £083.5 

CO 

3709.0 

1095.7 

50 

4463.1 

1533.1 

50 

5327.4  ;  2C94.1 

66 

3720.9 

1102.2 

76 

4476.5 

1541.4 

86 

5343.0  i  2104.7 

10 

3732.7 

1108.6     ' 

10 

4489.9 

1549.7 

10 

5358.6      2115.3 

20 

3744.6 

1115.1 

20 

4503.4 

1558.0 

20 

5S74.2     2126.0 

80 

3756.5 

1121.7 

30 

4516.9 

1566.3 

30 

53KL9  1  2136.7 

40 

3768.5 

1128.2 

40 

4530.4 

1574.7 

40 

5405.6     2147.5 

50 

3780.4 

1134.8 

50 

4544.0      1583.1 

CO 

5421.4  I  2158.4 

67 

3792.4 

1141.4         77 

4557.6     1591.6 

87 

5437.2     2169.2 

10 

3804.4 

1148.0     !            10 

4571.2 

1600.1 

10 

5453.1      2180.2 

20 

3816.4      1154.7     ;           20 

4584.8 

1608.6              20 

5469.0 

2191.1 

30 

3828.4     1161.3 

30 

4598.5 

1617.1 

30 

5484.9      2202.2 

40 

3840.5      1168.1 

40 

4612.2 

1625.7 

40 

5500.9 

2213.2 

50 

3852.6 

1174.8 

50 

4626.0 

1634.4 

50 

5517.0 

2224.3 

68            3864.7 

1181.6 

78 

4639.8 

1643.0 

88 

5533.1 

2235.5 

10 

3873.8 

1188.4               10 

4653.6 

1651.7 

10 

5549.2     2246.7 

20 

3889.0 

1195.2      i         -20 

4667.4 

1660.5 

20 

5565.4      2258.0 

30 

3901.2      1202.0     ||          30 

4681.3 

1669.2 

30 

5581.6      2269.3 

40 

3913.4      1208.9                40 

4695.2 

1678.1 

40 

5597.8     2280.6 

50 

3925.6 

1215.8 

50 

4709.2 

1686.9 

50 

5614.2     2292.0 

69 

3937.9 

1222.7 

79 

4723.2 

1695.8 

89 

5630.5     2£03.5 

10 

3950.2 

1229.7 

10 

4737.2 

1704.7 

10 

£646.9      2315.0 

20  I  3962.5      1236.7 

20 

4751  .2 

1713.7 

20 

5663.4     2326.6 

30  I  3974.8 

1243.7                30 

4765.3 

1722.7 

30 

£679.9     2338.2 

40     3987.2     12508    1           40 

4779.4 

1731.7 

40 

56964     2349.8 

50     3999.5 

1257.9      I          50 

4793.6 

1740.8 

50 

5713.0.     2361.5 

70 

4011.9 

1265.0     1    80 

4807.7 

1749.9 

90 

6729.7 

2373  3 

10     4024.4      1272.1     1            10 

4822.0 

1759.0 

10 

5746.3  i  2385.1 

20     4036.8 

1279.3      i          20 

4836.2 

1768.2 

20 

5763.1   .  2S97.0 

30     40493      1286.5     |i          30 

4850.5 

1777.4 

30 

5779  9  !  2408.9 

40 

4061.8 

1293.6      !          40 

4864.8 

1786.7 

40 

5796.7  i  2420.9 

50  1  4074.4 

1300.9               50     4879.2     1796.0             50 

5813.6  j  2432.9 

.291 


:AELE  VI.-TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 

A 

Tan- 
gent. 

T. 

Ex- 
ternal. 

E. 

Angle. 

A 

Tan-    ' 
gent. 

T. 

temaY   '  Angle. 

E.                     A 

Tan- 
gent. 

T. 

Ex- 
ternal 
E. 

91° 

5830.5 

2444.9 

101°         6950.6  i  3278.1       111°         8336.7 

4386.1 

10' 

5847.5 

2457.1 

10'  j  0971.3  i  3294.1  n            10' 

8362.7 

4407.6 

20 

5864.6 

2469.3 

20     6992.0  i  3310.1                20 

8388.9      4429.2 

30 

5881.7 

2481.5 

30     7012.7  ;  3326.1                30 

8415.1 

4450.9 

40 

5898.8 

2493.8 

40  i  7033.6  ;  3342.3  i             40 

8441.5 

4472.7 

50 

5916.0 

2506.1 

50  1-7054.5      3358.5    i            50 

8468.0     4494.6 

92 

5933.2 

2518.5 

102           7075.5      3374.9       112 

8-194.6 

4516.6 

10 

5950.5 

2531.0  I              10      7096.6      3391.2  i             10 

8521.3 

4538.8 

20     5067.9 

2543.5  t             20     7117.8      3407.7                20 

8548.1 

4561.1 

30     5985.3 

2556.0 

30  !  7139.0     3424.3                30 

8575.0     4583.4 

40 

6002.7 

2568.6 

40 

7160.3      3440.9 

40 

8602.1      4606.0 

50 

6020.2 

2581.3 

50 

7181.7     3457.6 

50 

8629.3  |  4628.6 

93 

6037.8 

2594.0 

103 

7203.2     3474.4 

113 

8656.6 

4651.3 

10 

6055.4 

260(5.8 

10      7224.7      3491.3 

10 

8684.0 

467'4.2 

20 

6073.1 

2019.7 

20      7246.3     3508.2 

20  1  8711.5 

4697.2 

30 

6030.8 

2632.6 

30     7268.0     3525.2 

30     8739.2 

4720.3 

40 

6108.6 

2645.5 

40  i  7289.8  i  &542.4 

40 

8767.0 

4743.6 

50 

6126.4 

2658.5    1            50  i  7311.7 

3559.6  i 

50 

8794.9 

4766.9 

94 

6144.3 

2671.6 

104 

7333.6 

3576.8 

114 

8822.9 

4790.4 

10 

6162.2 

2684.7 

10 

7355.6 

3594.2 

10 

8851.0 

4814.1 

20 

6180.2 

2697.9  1             20     7377.8 

3611.7 

20 

8879.3 

4837.8 

30 

6198.3 

2711.2    1            30      7399.9 

3629.2 

30 

8907.7 

4861.7 

40 

6210.4 

2724.5 

40     7422.2 

3646.8 

40 

8936.3 

4885.7 

50 

6234.6 

2737.9 

50     7444.6 

3664.5 

50 

8965  0 

4909.9 

95 

6252.8 

2751.3 

105 

7467.0     3682.3 

115 

8993.8 

4934.1 

10 

6271.1 

2764.8 

10     7489.6     3700.2 

10 

9022.7     4958.6 

20 

6289.4 

2778.3 

20      7512.2 

3718.2 

20 

9051.7  !  4983.1 

30 

6307.9 

2792.0 

30  |  7534.9     3736.2 

30 

9080.9  I  5007.8 

40 

6326.3 

2305.6 

40      7557.7     3754.4 

40 

9110.3 

5032.6 

50 

6344.8 

2819.4 

50 

7580.5 

3772.6 

50 

9139.8 

5057.6 

98 

6353.4 

2333.2 

106 

7603.5 

3791.0       116 

9169.4 

5082.7 

10 

6382.1 

2847.0 

10 

7626.6 

3809.4                10 

9199.1 

5107.9 

20     0400.8 

2861.0 

20 

7649.7 

3827.9 

20 

9229.0 

5133.3 

30  i  6419.5 

2875.0 

30     7672.9 

3846.5 

30 

9259.0 

5158.8 

40     643S.4 

2889.0 

40  i  7096.3" 

3865.2 

40 

9289.2 

5184.5 

50 

6457.3 

2903.1 

.   50 

7719.7 

3884.0 

CO 

9319.5 

5210.3 

97 

6476.2 

2917.3 

107 

7743.2 

3902.9 

117 

9349.9 

5236.2 

10     0495.2 

2931.6 

10      7766.8 

3921.9 

10 

9380.5 

5262.3 

20      6514.3 

2945.9 

20 

7790.5 

3940.9 

20 

9411.3 

5288.6 

30     6533.4 

2960.3 

30 

7814.3 

3960.1 

30 

9442.2 

5315.0 

40     655:2.6 

2974.7 

40 

7838.1 

3979.4 

40 

9473.2 

5341.5 

50 

6571.9 

2989.2 

50 

7862.1 

3998.7 

50 

9504.4 

5368.2 

98 

6591.2 

3003.8 

108 

7886.2 

4018.2 

118 

9535.7 

5895.1 

10     6610.6 

3018.4 

10 

7910.4 

4037.8 

10 

9567.2 

5422.1 

20      6630.1 

3033.1 

20 

7934.6 

4057.4 

20 

9598.9 

5449.2 

30 

6649.6 

3047.9 

30 

7959.0 

4077.2 

30 

9630.7 

5476.5 

40 

6669.2 

3062.8 

40 

7983.5 

4097.1 

40 

9662.6 

5504.0 

50 

6688.8 

3077.7 

50 

8008  0 

4117.0 

50 

9694.7 

5531.7 

99 

6708.6 

3092.7 

109 

8032.7 

4137.1 

119 

9727.0 

5559.4 

10 

0723.4 

3107.7 

10 

8057.4 

4157.3 

10 

9759.4 

5587.4 

20 

6748.2 

3122.9 

20 

8082.3     4177.5 

20 

9792.0 

5615.5 

30  !  6768.1 

3138.1 

30     8107.3     4197.9 

30 

9824.8 

5643.8 

40 

6788.1 

3153.3 

40      8132.3 

4218.4 

40 

9857.7 

567'2.3 

50 

6808.2 

3168.7 

50  ;  8157  5 

4239.0 

50 

9890.8 

57v0.9 

100 

6828.3 

3184.1 

110        i  8182.8 

4259.7 

120 

9924.0 

5729.7 

10 

6848.5 

!  3199.6 

10     8208.2 

4280.5 

10 

9957.5      5758.6 

20 

6868.8 

3215.1 

20 

8233.7 

4301.4 

20     9991.0  ;  5787.7 

30 

6889.2 

3230.8 

30 

8259.3 

4322.4 

30 

10025.0  !  5817.0 

40 

6909.6 

3246.5 

40 

8285.0 

^1343.6 

40 

10059.0     5846.5 

50 

6930.1 

3262.3 

50 

8310.8 

4364.8 

50 

10093.0     5876.1 

TABLE  VII.— LONG  CHORDS. 


Degree 

0£' 

Curve. 

Actual 
Are, 
One 
Station. 

LONG  CHORDS. 

2                    3 

Stations.      Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

0°  10' 

100.000 

200.000 

299.999 

399998         499.996 

599.993 

20 

.000 

199.999     1      299.997 

899.992 

499.983 

599.970 

30 

.000 

199.998     |      299.992 

399.981 

499.962 

599.933 

40 

.001 

199.997     I       299.986 

399.966 

499.932 

599.882 

50 

.001 

199.995     |       299.979 

399.947         499.894 

599.815 

1 

100.001 

199.992     |       299.970 

399.924 

499.848 

599.733 

10 

.002 

199.990 

299.959 

399.89(5 

499.793 

599.637 

20 

.002 

199.986 

299.946 

399.865 

499.729 

599.526 

30 

.003 

199.  983 

299.932 

399.829 

499.657 

599.401 

40 

.003 

199.979 

299.915 

899.789 

499.577 

599.260 

50 

.004 

199.974 

299.898 

399.744 

499.488 

599.105 

2 

100.005 

199.970 

299.878 

399.695 

499.391 

598.934 

10 

.000 

199.964 

299.857 

399.643 

499.285 

598.750 

8@ 

.007 

199.959 

299.834 

399.586 

499.171 

598.550 

30 

.008 

199.952 

299.810 

399.524 

499.049 

598.  336 

40 

.009 

199.946 

299.783 

399.459 

498.918 

598.106 

50 

.010 

199.939 

299.756 

399.389 

498.778 

597.8(52 

3 

100.011 

199.931 

299.726 

399.315 

498.6:30 

597.604 

10 

.013 

199.924 

299.695 

399.237 

498.474 

597.331 

20 

.014 

199.915 

299.662 

399.154 

-498.309 

597.043 

yo 

.015 

199.907 

299.627 

399.068 

498.136 

596.740 

40 

•017 

199.898 

299.591 

398.977 

497.955 

596.423 

50 

.019 

199.888 

299.553 

398.882 

497.765 

596.091 

4 

100.020 

199.878 

299.513 

398.782 

497.566 

595.744 

10 

.022 

199.868 

299.471 

398.679 

497.360 

595.383 

20 

.024 

199.857 

299.428 

398.571 

497.145 

595.007 

bO 

.026 

199.846 

299.383 

398.459 

496.921 

594.617 

40 

.028 

199.834 

299.337 

398.343 

496.689 

594.212 

50 

.030 

199.822           299.289 

39S.223 

496.449 

593.792 

5 

100.032 

199.810 

299.239 

398.099 

496.201 

593.358 

10 

.034 

199.797 

299.187 

397.970 

495.944 

592.909 

20 

.036 

199.783 

299.134 

397.837 

495.678 

592.446 

30 

.0=38 

199.770 

299.079 

397.700 

495.405 

591.968 

40 

.041 

199.756 

299.023 

397.559 

495.123 

591.476 

50 

.043 

199.741 

298.964 

397.413 

494.832 

590.970 

6 

100.046 

199.726 

298.904 

397.264 

494.534 

590.449 

10 

.048 

199.710 

298.843 

397.110 

494.227 

589.913 

20 

.051 

199.695 

298.779 

396.952 

493.912 

589.364 

30 

.054 

199.678 

298.714 

396.790 

493.588 

588.800 

40 

.056 

199.662 

298.648 

396.623 

493.257 

588.221 

50 

.059 

199.644 

298.579 

396.453 

492.917 

587.628 

7 

100.062 

199.627 

298.509 

396.278 

492.568 

587.021 

10 

.065 

199.609 

298.438 

396.099 

492.212 

586.400 

20 

.068 

199.591 

298.364 

395.916 

491.847 

585.765 

30 

.071 

199.572 

298.289 

393.729 

491.474 

585.115 

40 

.075 

199.553 

298.212 

395.538 

491.093 

584.451 

50 

.078 

199.533 

298.134 

395.342 

490.704 

583.773 

8 

100.081 

199.513 

298.054 

395.142 

490.306 

583.081 

10 

.085 

199.492 

297.972 

394.9:38     !     489.900 

582.375 

20 

.088 

199.471 

297.888 

394.731     !    489.486 

581.654 

30 

.092 

199.450 

297.803 

394.518 

489.064 

580.920 

40 

.095 

199.428 

297.716 

394.302 

488.634         580.172 

5-J 

.099 

199.406 

297.628 

394.082 

488.196         579.409 

9 

100.103 

199.383 

297.538         393.857 

487.749 

578.633 

10 

.107 

199.360 

297.446         393.629 

487.294 

577.843 

20 

.111 

199.337 

297.352 

393.396 

486.832 

577.039 

30 

.115 

199.313 

297.257         393.159 

486.361 

576.222 

40 

.119 

199.289 

297.160         392.918 

485.882 

575.390 

50 

.123 

199.264 

297.062         392.673 

485.395 

574.545 

10 

100.127 

199.239 

296.962         392.424 

484.900 

573.686 

-   298 


TABLE  VII.-LONG  CHORDS. 


Degree 
of 
Curve. 

LONG  CHORDS. 

7 
Stations. 

8 

Stations. 

Stations. 

10 

Stations. 

Stations. 

12 

Stations. 

0°10'        699.988 

799.982    1      899.974 

999.965 

1099.95 

1199.94 

20         699.953 

799.929           899.899 

999.860 

1099.81 

1199.76 

30         099.893 

799.840           899.772 

999.686 

1099.58 

1199.46 

40         699.810 

799  716 

899.594 

999.442 

1099.25 

1199.03 

50         699.704 

799.550 

899.365 

999.128 

1098.84 

1198.49 

1                699.574 

799.360 

899.086         998.744 

1098.33 

1197.82 

10         699.420 

799.130 

898.757         998.290 

1097.72 

1O7.04 

20         699.242 

798.863 

898.376 

997.708 

1097.02 

11JJ6.13 

30         699.041 

798.562 

897.945 

997.175 

1096.23 

1195.11 

40         698.816 

798.224 

897.464 

996.513 

1095.35 

1193.90 

50    1    698.567 

797.852 

896.931 

995.782 

1094.38 

1192.09 

2           j     698.295 

797.444 

896.349 

994.981 

1093.31 

1191.31 

10 

698.000 

797  000 

895.716 

994.112 

1092.15 

11MJ.80 

20 

697.080           790.522 

895.033 

993.173 

1090.90 

lit:  8.  18 

30 

607.838           796.008 

894.299 

992.165 

1089.56 

1180.43 

40 

696.971 

795.459 

893.515 

991.088 

1088.12 

1184.57 

50 

090.581 

794.874 

892.681 

989.943 

10S6.60 

11^-2.59 

3 

690.108 

794.255 

891.798 

988.729 

1084.98 

1180.49 

10 

095.731 

793.600 

890.864 

987.447 

1083.28 

1178.28 

20 

095.271 

792.911 

889.880 

986.096 

1081.48 

1175.94 

30 

094.787 

792.186 

888.846 

984.677 

1079.59 

1173.49 

40 

094.280 

791.427 

887.703 

983.190 

1077.61 

1170.93 

50 

093.750 

790.032 

886.630 

981.036 

1075.54 

1168.25 

4 

693.196 

789.803 

885.448 

980.014 

1073.38 

1105.45 

10 

608.619 

788.939 

884.217 

978.325 

1071.14 

1102.54 

20 

692.018 

788.040 

882.936 

970.509 

10B8.81 

1159.51 

30 

091.395 

787.108 

881.606 

974.746 

1006.38 

1156.37 

40 

690.748 

786.140 

880.228     i     972.856 

1003.87 

1153.12 

50 

090.079 

785.138 

878.800 

970.900 

1061.27 

1149.76 

5 

089.380 

784.101 

877.324 

968.  877 

1058.59 

1140.28 

10 

688.070 

783.030 

875.800 

966.788 

1055.81 

1142.09 

20 

687.930 

781.925 

874.227 

964.634 

1052.95 

1138.99 

30 

687.169 

780.786 

872.605 

962.415 

1050.01 

1135.18 

40 

080.384 

779.012 

870.936 

960.130 

1046.97 

1131.26 

50 

Oa5.576 

778.406 

869.219 

957.780 

1043.86 

1127.24 

6 

684.745 

777.165 

867.454 

955.366 

1040.66 

1123.10 

10 

083.892 

775.890 

865.642 

952.888 

1037.37 

1118.86 

20 

083.016 

774.582 

803.782 

950.345 

1034.01 

1114.51 

30 

082.117 

773.240 

801.875 

947.7'39         1030.55 

1110.05 

40 

081  .  195 

771.804 

859.922 

945.069 

1027.02 

1105.49 

50 

680.251 

770.455 

857.921 

942.337 

1023.40 

1100.83 

7 

079.285 

769.014 

a55.874 

939.542         1019.70 

1096.  CG 

10     :     678.296 

707.539 

853.780 

936.084         1015.93 

1091.19 

20     i     677.284 

706.030 

851.640 

933.764         1012.07 

1086.22 

30         676.250 

764.490 

849.455 

930.783         1008.13 

1081.15 

40     ;     675.194 

762.916 

847.224 

927.7'41          1004.11 

1075.98 

50     ;     07'4.116 

761.309 

844.947 

924.038         1000.01 

1070.71 

8 

673.015 

759.670 

842.625 

921.474          -095.834  |     1065.34 

10 

071.892 

757.999 

840.258 

918.250 

991.580  |     1059.88 

20         670.748 

756.295 

837.845 

914.906 

987.250       1054.32 

30 

069.581 

754.560 

835.389 

911.023 

982.844        1048.00 

40 

668.393 

752.792 

832.888 

908.221 

978.302       1042.91 

50 

667.182 

750.993 

830.342 

904.761           973.806       1037.00 

9 

605.950 

749.161 

827.75-1         901.242           969.175       1031.13 

10 

064.697 

747.299 

825.121         897.667           964.471        1025.11 

20 

663.421 

745.404 

822.445 

894.  033           959.094       1018.99 

30 

662.124 

743.479 

819.726 

890.343           954.844 

1012.79 

40 

660.806 

741.522 

816.965         886.597           949.924 

1006.49 

50 

659.406 

739.535 

814.160         882.795           944.933 

1000.12 

10 

658.105 

787.516 

811.314         878.938           939.871 

993.653 

294 


TABLE  VII.— LONG  CHORDS. 


Degree 
of 
Curve. 

Actual 
Arc, 
One 
Station. 

LONG  CHORDS. 

2 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations.    \ 

10°  10'      100.131 

199.213 

296.860 

392.171 

484.397         572.813 

20 

.136 

199.187 

296.756 

391.914 

483.886 

571.926 

30             .140 

199.161 

296.651         391.652 

483.367 

571  .027 

40             .145 

199.134 

296.544     ,     391.387 

482.840         570.113 

50             .149 

199.107 

296.436         391.117 

482.305     i     569.  1MB 

11              100.154 

199.079 

296.325         390.843 

481.762         568.245 

10  i           .158 

199.051 

296.214         390.565 

481.211 

567.292 

20 

.163 

199.023 

296.100 

390.284 

480.653 

566.324 

30 

.168 

198.994 

295.985         389.998 

480.086 

565.343 

40 

.173 

198.964 

295.868 

389.708 

479.511 

564.349 

50 

.178 

198.935 

295.750 

389.414 

478.929 

563.341 

12 

100.183 

198.904 

295.629 

389.116 

478.338 

562.321 

10 

.188 

198.874 

295.508         388.814 

477.740 

561.287 

20 

.193 

198.843 

295.384         388.508 

477.135 

560.240 

30 

.199           198.  8il 

295.259     ;     388.197 

476.521 

559.180 

40 

.204           198.779 

295.132     '     387.883         475.899 

558.107 

50 

.209           198.747 

295.004         387.565         475.270 

557.020 

13 

100.215  1         19S.714 

294.874         387.243 

474.633 

555.921 

10 

.220           198.081 

294.742         386.916 

473.988 

554.809 

20 

-226           198.648 

294.609         386.586 

-473.336 

553.684 

30  ;           .232  i         198.614             294.474         38(5.252 

472.675 

552.546 

40             .237  !        198.579            294.337         385.914 

472.007 

551.395 

50             .213           198.544 

294.199         385.572 

471.332         550.232 

14 

100.249  I         198.509 

294.059     !     385.225 

470.649 

549.056 

10 

.255  1        198.474 

293.918         384.87'5 

469.958 

547.867 

20 

.261  i         198.437 

293.774     \     384.521 

469.260 

546.666 

30 

.267  i         198.401 

293.629     '     384.163         468.554 

545.452 

40 

.274  i         198.364 

293.483     !     383.801     1     467.840     '     544.226 

50 

.280  i        198.327 

293.335 

383.4;i5 

467.119     !     542.987 

15 

100.286  i        198.289 

293.185         383.065 

466.390 

541.736 

10 

•  292 

198.251 

293.034         382.691 

465.6.54 

540.472 

20 

•  299 

198.212 

292.881     !     382.313 

464.911 

539.196 

30 

.306 

198.173 

292.726     !     381.931 

464.160 

537.908 

40 

.312 

198.134 

292.570     |     381.546 

463.401 

536.608 

50 

.319 

198.094 

292.412         381.156 

462.635 

535.296 

is: 

100.326 

198.054 

292.252 

380.763 

461.862 

533.972 

10 

.333 

198.013 

292.091 

380.365 

461.081 

532.635 

20 

.339 

197.972 

291.928 

379.964 

460.293 

531.287 

30 

.346 

197.930 

291.764 

379.559 

459.498 

529.927 

40 

.353 

197.888 

291.598 

379.150 

458.695 

528.555 

50 

.361 

197.846 

291.430 

378.737 

457.886 

527.171 

17 

100.368 

197.803 

291.261 

378.320         457.069 

525.778 

10 

.375 

197.760 

291.090 

377.900 

456.244 

524.369 

20 

.382 

197.716 

290.918 

377.475 

455.413 

522.950 

30 

.390 

197.672 

290.743 

377.047 

454.574 

521.519 

40 

.397 

197.628 

290.568 

376.615 

453.728 

520.078 

50 

.405 

197.583 

290.390 

376.179 

452.875 

518.625 

18 

100.412 

197.538 

290.211 

375.739 

452.015 

517.160 

10 

.420 

197.492            290.031 

375.205         451.147 

515.685 

20-             .428 

197.446 

289.849 

374.848         450.373 

514.198 

30  !           .436 

197.399 

289.665         374.397     j     449.392 

512.699 

40             .444 

197.352 

289.479         373.942 

448.504 

511.190 

50             .452 

197.305 

289.292         373.483 

447.608 

509.H70 

19              100.460 

197.256 

289.104         373.021 

446.706 

508.139 

10             .468 

197.209 

288.913     i     372.554 

445.797 

506.597 

20 

.476           197.160 

288.722     i     372.084 

444.881 

505.043 

30 

.484  j        197.111 

288.528         371.610 

443.957 

503.479 

40 

.493           197.062 

288.333         871.133 

443.028 

501.905 

50 

.501           197.012 

288.137         370.652     i     442.091 

500.320 

20             100.510           196.962            287.939    i     370.167         441.147 

498.724 

295 


TABLE  VDL— LONG   CHORDS. 


Degree 
of 
Curve. 

LONG  CHORDS. 

7 
Stations. 

8 
Stations. 

9                    10 

Stations.       Stations. 

11 

Stations. 

12 

Stations. 

i 
10°  10'      656.  723 

735.467 

808.426 

875.025 

934.741         987.105 

20 

655.320 

733.387 

805.495 

871.058 

929.542         980.473 

30 

653.895 

731.277 

802.524 

867.038 

924.276         973.760 

40 

652.450 

729.137 

799.512 

862.963 

918.943 

966.967 

50 

650.  983 

726.967 

796.458 

858.836 

913.544 

960.093 

11 

649.496 

724.767 

793.364 

854.656 

908.080 

953.141 

10 

647.989 

722.537 

790.230 

850.425 

902.550 

946.112 

20 

646.460 

720.278 

787.056 

846.140 

896.957         939.007 

30 

644.911 

717.990 

783.843 

841.808 

891.303     i     931.828 

40 

643.342 

715.672 

780.590 

837.424 

885.586     !     924.575 

50 

641.752 

713.325 

777.298 

832.990 

879.807         917.250 

12 

640.142 

710.950 

773.968 

828.507 

873.968 

9C9.&54 

10 

638.512 

708.546 

770.600 

823.974 

868.070 

902.389 

20 

636.862 

706.113 

767.193 

819.394 

862.113 

894.855 

30 

635.191 

703.653 

763.749 

814.766 

856.099         887.254 

40 

6:33.501 

7'01.164 

760.268 

810.092 

850.028         879.588 

50 

631.792 

698.647 

756.749 

805.370 

843.900         871.857 

13 

630.062 

696.103 

753.194 

800.602 

837.718 

864.063 

10 

628.313 

693.531 

749.603 

795.790 

831.482 

856.208 

20 

626.544 

690.932 

745.976 

790.932 

825.192 

848.293 

30 

624.756 

688.306 

742.313 

786.030 

818.850 

840.318 

40 

622.949 

685.653 

738.616 

781.065 

812.457 

832.286 

50 

621.123 

682.974 

734.883 

776.096 

806.013 

824.198 

14 

619.278 

680.268 

731.116 

771.066 

799.520 

816.056 

10 

617.413 

677.535 

727.315 

765.993 

792.979 

807.860 

20 

615.530 

674.777 

723.480 

760.879 

786.389 

799.612 

30 

613.628 

671.993 

719.612 

755.725 

779.753 

791.313 

40 

611.708 

669.183 

715.711 

750.531 

773.072 

782.966 

50 

609.769 

666.348 

711.777 

745.297 

766.345 

774.571 

15 

607.812 

663.488 

707.811 

740.024 

759.575 

766.130 

10 

605.836 

660.603 

703.814 

734.714 

752.763 

20 

603.842 

657.693 

699.785 

729.366 

745.908 

30 

601,881 

654.758 

695.725 

723.982 

739.014 

40 

599.801 

651.799 

691.634 

718.561 

732.078 

50 

597.753 

648.817 

687.513 

713.105 

725.104 

16 

595.688 

645.810 

683.362 

707.614 

718.092 

10 

593.605 

642.780 

679.182 

702.088 

711.043 

20 

591.505 

639.727 

674.973 

696.529 

703.959 

30 

589.388 

636.650 

670.735 

690.938 

40 

587.253 

633.550 

666.469 

685.314 

50 

585.101 

630.428 

662.175 

679.659 

17 

582.933 

627.283 

657.854 

.673.972 

10 

580.747 

624.117 

653.506 

668.256 

20 

578.545 

620.928 

649.131 

662.510 

30 

576.326 

617.717 

644.730 

656.735 

40 

574.091 

614.485 

640.304 

650.933 

50 

571.839 

611.232 

635  .  852 

645.103 

18 

569.571 

607.958 

631.375 

639.245 

10 

567.287 

604.664 

626.874 

• 

20 

564.988 

601.349 

622.349 

30 

562.673 

598.013 

617.801 

40 

560.342 

594.658 

613.229 

50 

557.996 

591.283 

608.635 

19 

555.634 

587.888 

604.018 

10 

553.257 

584.475 

599.379 

20 

550.864 

581.012 

594.720 

30 

548.457 

577.591 

590.039 

40 

546.  035 

574.121 

585.339 

50 

543.599 

570.634 

580.618 

20 

541.147  i        567.128            575.877 

1 

296 


TABLE  VII.— LONG  CHORDS. 


LONG  CHORDS. 

Degree 

of 

Actual 
Arc, 

[!>UTV6 

One 

3 

3 

4 

5 

6 

Station. 

Stations. 

Stations. 

Stations. 

Stations. 

Stations. 

21° 

100.562 

196.651 

286.716 

367.179 

435.845 

488.931 

22 

100.617 

196.325 

285.  437 

364.060 

429.305 

478.775 

23 

100.675 

195.985 

284.101 

3(50.810 

423.033 

468.270 

24 

100.735           195.630 

282.709 

357.433 

416.535         457.433 

25 

100.798           195.259 

281.262 

353.930 

489,819 

446.280 

20 

100.863           194.874 

279.759 

350.303 

402.891     1     434.827 

27 

100.931           194.474 

278.201 

346.555 

395.758 

423.092 

28 

101.002 

194.059 

276.589 

342.688 

388.428 

411.092 

29 

101.075 

193.630 

274.924 

338.704 

380.908 

398.846 

30 

101.152 

193.185 

273.205 

334.607 

373.205 

386.370 

i 

297 


TABLE  VIII  -MIDDLE  ORDIMTES. 


TABLE  VIII. -MIDDLE  ORDIXATE3. 


Degree 

of 
Curve. 

1 

Station. 

2 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

j  • 

0°  Itf 

.036 

,145 

.32? 

.582 

.909 

1.309 

20 

.073 

.291 

.654 

1.164 

1.818 

2.618 

30 

.109 

.436 

.982 

1.745 

2.727 

3.926 

40 

.145 

.582 

1.309 

2.327 

3.  036 

5.235 

50 

.182 

.727 

1.636 

2.909 

4.545 

6.544 

1 

.218 

.873 

1.963 

3.490 

5.453 

7.852 

10 

.255 

1.018 

2.291 

4.072 

6.362 

9.160 

20 

.291 

1.164 

2.618 

4.654 

7.270 

10.468 

30 

.327 

1.309 

2.945 

5.235 

8.179 

11.775 

40 

.364 

1.454 

3.272 

5.816 

9.087 

13.082 

50 

.400 

1.600 

3.599 

6.393 

9.994 

14.389 

2 

.430 

1.745 

3.926 

6.979 

10.902 

15.694 

10 

.473 

1.891 

4.253 

7.560 

11.809 

17.000 

20 

.509 

2.036 

4.580 

8.141 

12.716 

18.304 

30 

.545 

2.181 

4.907 

8.722 

13.623 

19.608 

40 

.582 

2.327 

5.234 

9.303 

14.529 

20.912 

50 

.618 

2.472 

5.561 

9.  £83 

15.485 

22.214 

3 

.654 

2.618 

5.888 

10.464 

16.341 

23.516 

10 

.691 

2.763 

6.215 

11.044 

17.246 

24.817 

20 

.727 

2.908 

6.542 

11.624    v 

18.151 

26.117 

30 

.763 

3.054 

6.868 

12.204 

19.055 

27.416 

40 

.800 

3.199 

7.195 

12.784 

19.959 

28.714 

50 

.836 

3.345 

7.522 

13.303 

20.863 

30.012 

4 

.872 

3.490 

7.848 

13.943 

21.766 

31.308 

10 

.909 

3.635 

8.175 

14.522 

22.668 

32.603 

20 

.945 

3.781 

8.501 

15.101 

23.570 

,33.896 

30 

.98-2 

3.926 

8.828 

15.680 

24.471 

35.189 

40 

1.018 

4.071 

9.154 

16.258 

25.372 

36.480 

50 

1.054 

4.217 

9.480 

16.837 

26.272 

37.770 

5 

1.091 

4.362 

9.807 

17.415 

27.171 

39.059 

10 

1.127 

4.507 

10.133 

17.992 

28.070 

40.346 

20 

1.164 

4.653 

10.459 

18.570 

28.968 

41.631 

30 

1.200 

4.798 

10.785 

19.147 

29.666 

42.916 

40 

1.237 

4.943 

11.111 

19.724 

30.762 

44.198 

50 

1.273 

5.038 

11.436 

20.301 

31.658 

45.479 

6 

1.303 

5.234 

11.762 

20.877 

32.553 

46.759 

10 

1.346 

5.379 

12.088 

21.453 

33.448 

48.037 

20 

1.332 

5.524 

12.413 

22.029 

34.341 

49.313 

30 

1.418 

5.669 

12.7'39 

22.604 

35.234 

50.587 

40 

1.45.") 

5.814 

13.064 

23.179 

36.126 

51.860 

50 

1.491 

5.960 

13.389 

23.754 

37.017 

53.130 

7 

1.528 

6.105 

13.715 

24.328 

37".  907 

54.399 

10 

1.564 

6.250 

14.040 

24.902 

38.796 

55.666 

20 

1.600 

6.395 

14.365 

25.476 

39.684 

56.931 

30 

1.G37 

6.540 

14.689 

26.049 

40.571 

58.193 

40 

1.673 

6.685 

15.014 

26.622 

41.458 

59.45-4 

50 

1.710 

6.831 

15.339 

27.195 

42.343 

60.712 

8 

1.746 

6.976 

15.663 

27.767 

43.227 

61.969 

10 

1.71:2 

7.121 

15.988 

28.338 

44.110 

63.223 

20 

1.819 

7.266 

16.312 

28.910 

44.992 

64.475 

30 

1.855 

7.411 

16.636 

29.481 

45.873 

65.724 

40 

1.892 

7.556 

16.960 

30.051 

46.753 

66.97'2 

50 

1.923 

7.701 

17.284 

30.621 

47.632 

68.216 

9 

1.965 

7.846 

17.608 

31.190 

48.510 

69.459 

10 

2.001 

7.991 

17.932 

31.759 

49.386 

70.699 

20 

2.037 

8.136 

18.255 

32.328 

50.261 

71.936 

30 

2.074 

8.281 

18.578 

32.896 

51.135 

73.171 

40 

2.110 

8.426 

18.902 

33.464 

52.008 

74  403 

50 

2.147 

8.571 

19.225 

34.031 

52.880 

75.632 

10 

2.183 

8.716 

19.548 

34.597 

53.750 

7'6.859 

298 


TABLE  VIII.— MIDDLE  ORDINATES. 


Degree 
of 
Curve. 

7 

Stations. 

8 

Stations. 

9 

Stations. 

1O 

Stations, 

11 

Stations. 

Stations. 

0°  10' 

1.782 

2.327 

2.945 

3.636 

4.400 

5.236 

20 

8.668 

4.654 

5.890 

7.272 

8.799 

10.471 

30 

5.S45 

6.981 

8.835 

10.907 

13.197 

15.704 

40 

7.126 

9.307 

11.778 

14.540 

17.593 

20.936 

50 

8.907 

11.632 

14.721 

18.173 

21.987 

26.164 

1 

10.687 

13.957 

17.663 

21,803 

26.378 

31.388 

10 

12.467 

16.281 

20.603 

25.431 

30.766 

36.607 

20 

14.246 

18.604 

23.541 

29.057 

35.150 

41.821 

30 

16.024 

20.925 

26.477 

32.679 

39.530 

47.028 

40 

17.802 

23.246 

29.411 

36.298 

43.905 

52.229 

50 

19.579 

25.564 

32.343 

39.914 

48.274 

57.422 

2 

21.355 

27.881 

35.272 

43.525 

52.637 

62.606 

10 

23.130 

30.197 

38.198 

47.131 

56.993 

67.780 

20 

24.903 

32.510 

41.121 

50.733 

61.343 

72.945 

30 

26.676 

34.821 

44.040 

54.330 

65.684 

78.098 

40 

28.447           37.K30 

46.956 

57.921 

70.018 

83.240 

50 

30.21(5 

39.436 

49.868 

61.506 

74.342 

88.370 

3 

31.984 

41.740 

52.776 

65.084 

78.657 

93.486 

10 

33.751 

44.041 

65.679 

68.656 

82.963 

98.588 

20 

35.516 

46..  339 

58.577 

72.221 

87.258 

108.675 

30          37.279 

48.634 

61.471 

75.778 

91.542 

108.747 

40 

39.040 

50.926 

64.360 

79.328 

95.814 

113.803 

50 

40.800 

53.215 

67.243 

82.869 

100.075 

118.841 

4 

42.557 

55.500 

70.121 

86.402 

104.323 

123.862 

10 

44.312 

57.781 

7'2.992 

89.925 

108.558 

128.864 

20 

46.065 

60.059           75.858 

93.440 

112.779 

133.847 

30 

47.816 

62.3=33 

78.717 

96.945 

116.986 

138.810 

40 

49.564 

64.602 

81.570 

100.489 

121.178 

143.753 

50 

51.310 

66.868 

84.416 

103.924 

125.  a56 

148.674 

5 

53.053 

69.129 

87.255         107.397 

129.517 

153.572 

10 

54.794 

71.386 

90.087 

110.860 

133.663 

158.448 

20 

56.532 

73.638 

92.911 

114.311 

137.791 

163.300 

30 

58.267 

75.885 

95.728 

117.751 

141.903 

168.128 

40 

59.999 

78.127 

98.536 

121.178          145.997 

172.931 

50 

61.729 

80.364 

101.337 

124.593           150.072 

177.708 

6 

63.455 

82.596 

104.129 

127.995           154.129 

182.459 

10 

65.178 

84.822 

106.912 

131.384          158.  1C6 

187.182 

20 

66.898 

87.043 

109.686 

134.759           162.184 

191.878 

30 

68.615 

89.258 

112.452 

138.120          166.182 

196.545 

40 

70.328 

91.468 

115.298 

141.468           170.159 

201.183 

50 

72.037 

93.671 

117.954 

144.800 

174.114 

205.792 

7 

73.744 

95.868 

120.691 

148.118 

178.048 

210.370 

10 

75.446 

98.059 

123.417 

151.421 

181.960 

214.916 

20 

77.145 

100.244 

126.134 

154.708 

185.850 

219.431 

30 

78.840 

102.422 

128.840 

157.979 

189.716 

223.914 

40 

80.531 

104.594 

131.535 

161.234 

193.559 

228.363 

50 

82.218 

106.758 

134.219 

164.473 

197.377 

232.779 

8 

as.  901 

108.916 

136.893 

167.695 

201.171 

237.160 

10 

85.580 

111.067 

139.555 

170.899 

204.941 

241.507 

20 

87.254 

113.210 

142.205 

174.086 

208.685 

245.818 

30 

88.924 

115.346     i     144.844 

177.255 

212.403 

250.093 

40 

90.590 

117.475 

147.470 

180.407 

216.095 

254.331 

50 

92.252 

119.596 

150.085 

183.539 

219.760 

258.531 

9 

93.909 

121.709 

152.687 

186.653 

223.398 

262.694 

10 

95.561 

123.814 

155.277 

189.748 

227.008 

266.818 

20 

97.208 

125.911 

157.854 

192.824 

230.591 

270.904 

30 

98.851 

128.000 

160.417 

195.880 

234.145 

274.949 

40 

100.489 

130.081 

162.968 

198.916 

237.670 

278.955 

50 

102.122 

132.153 

165.505 

201.932 

241.167 

282.919 

10 

103.750 

134.217 

168.029 

204.928 

244.633 

286.843 

TABLE  VIII.— MIDDLE  ORDINATES. 


Degree 
of 
Curve. 

Station. 

2 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

10°  10' 

2.219 

8.860 

19.87'0 

35.164 

54.619 

7'8.083 

20 

2.256 

9.005 

20.15)3 

85.729 

55.486 

79.305 

30 

2.293 

9.150 

20.516 

36.294 

56.358 

80.523 

40 

2.329 

9.295 

20.888 

36.859 

57.218 

81.739 

50 

2.3C5 

9.440 

21  .  ICO 

37.428 

58.081 

82.951 

11 

2.402 

9.585 

21.483 

87.9HG 

58.943 

84.161 

10 

2.438 

9.729 

21.804 

38.549 

59.804 

85.368 

20 

2.475 

9.874 

22.126 

39.111 

60.603 

86.571 

30 

2.511 

10.019 

22.448 

39.673 

61.521 

87.772 

40 

2.547 

10.164 

22.769 

40.234 

62.S77 

88.969 

50 

2.584 

10.308 

23.090 

40.795 

63.232 

90.164 

12 

2.620 

10.453 

23.412 

41.355 

64.085 

91.355 

10 

2.657 

10.597 

23.732 

41.914 

64.937 

92.542 

20 

2.693 

10.742 

24.053 

42.473 

05.787 

93.727 

30 

2.730 

10.887 

24.374 

43.031 

66.636 

94.908 

40 

2.766 

11.031 

24.694 

43.588 

67.482 

96.086 

50 

2.803 

11.176 

25.014 

44.145 

68.328 

97.260 

13 

2.839 

11.320 

25.334 

44.701          69.171 

98.431 

10 

2.876 

11.405 

25.654 

45.256         70.013 

<  19.  5118 

20 

2.912 

11.609 

25.97'4 

45.811        .70.854 

100.702 

30 

2.949 

11.75-1 

26.293 

46.365         71.692 

101.922 

40 

2.985 

11.898 

26.612 

46.919     i     72.529 

108.079 

50 

3.022 

12.043 

26.931 

47.47'2 

73.864 

104.232 

14 

3.058 

12.187 

27.250 

48.024 

74.197 

105.381 

10 

3.095 

12.331 

27.569 

48.575         75.020 

106.527 

20 

•3.131 

12.476 

27.887 

49.126          7T>.K)9 

107.669 

30 

3.168 

12.620 

28.206 

49.676 

76.687 

108.807 

40 

3.204 

12.764 

28.524 

50.225 

77.513 

109.941 

5J 

3.241 

12.908 

28.841 

50.773 

7'8.337 

111.071 

15 

3.277 

13.053 

29.159 

51.321 

7'9.159 

112.197 

10 

3.314 

13.197 

29.476 

51.868 

79  .  U79 

113.319 

20 

3.350 

13.341 

29.794 

52.414 

80.7C8 

114.438 

30 

3.387 

13.485 

30.111 

52.959 

81.614 

115.  E52 

40 

3.423 

13.629 

30.427 

53.504 

82.429 

116.662 

50 

3.460 

13.773 

30.744 

54.048 

83.241 

117.7fc8 

1C 

3.496 

13.917 

31.060 

54.591 

84.052 

118.870 

10 

3.533 

14.061 

31.376 

55.183 

84.861 

119.5:67 

20 

3.569 

14.205 

31.692 

55.675 

85  .  667 

121.  C61 

30 

3.606 

14.349 

32.008 

56.215 

86.471 

122.150 

40 

3.643 

14.493 

32.323 

56.755 

87.274 

123.225 

50 

3.679 

14.637 

32.6:38 

57.294 

88.074 

124.315 

17 

3.716 

14.781 

32.953 

57.832 

88.872 

125.891 

10 

3.752 

14.925 

a3.  267 

58.369 

89.668 

126.463 

20 

3.789 

15.069 

33.582 

58.906 

90.462 

127.530 

30 

3.825 

15.212 

33.896 

59.441 

91.254 

128.593 

40 

3.862 

15.356 

34.210 

59.976 

92.043 

129.651 

50 

3.899 

15.500 

34.523 

60.510 

92.830 

180.704 

18 

3.935 

15.643 

34.837 

61.042 

93.616 

131.753 

10 

3.972 

15.787 

35.150 

61.574 

94.3«8 

132.797 

3D 

4.008 

15.931 

35.463 

62.106 

05.179 

133.837 

30 

4.045 

16.074 

35.775 

eav6S6 

95.957 

134.872 

40 

4.081 

16  218 

36.088 

63.165 

96.783 

135.  C02 

50 

4.118 

16.361 

36.400 

63.693 

117.5*  Hi 

136.928 

19 

4.155: 

16.505 

36.712 

64.221 

98  278 

137.948 

10 

4.191: 

16.648 

37.023 

64.747 

!!'.».  017 

138.  SC4 

20 

4.228 

16.792 

ar.834 

65.2?3 

90.818 

13(t!)75 

30 

4.265 

16  9.35 

37.645 

65.797 

100.577 

140.981 

40 

4.301 

17.078 

37.956 

66.321 

101.  339 

141.982 

50 

4.338 

17.222 

38.266.: 

66.843 

102.098 

142.978 

20 

4.374 

17.365.  :|      38.576: 

67.365 

102.  855 

143.969 

300 


TABLE  IX.— LINEAR  DEFLECTION  TABLE. 


Deflec- 
tion. 

100. 

200. 

300. 

400. 

500. 

600. 

700. 

ceo. 

900. 

1000. 

30' 

0.87      1.7'5 

2.62 

3.49 

4.30 

5.24 

6.11 

6.98 

7.85 

8.73 

1° 

1.75 

3.49      5.21 

6.98 

8.73 

10.47 

12.22 

13.96 

15.71 

17,  45 

3D 

2.03 

i     5.24 

7.8.3 

10.47 

13.09 

15.71 

18.33 

20.94 

23.56 

26.18 

2 

3  49 

'     6.98    10.47 

13.90 

17.45 

20.94 

24.43 

27.92 

31.41 

34.90 

30 

4.36 

!     8.73    13.  OJ 

17.45 

21.81 

26.18 

30.54     34.90 

39.27 

43.63 

3 

5.34 

10.47 

15.71 

20.94 

20.18 

31.41     36.65     41.88 

47.12 

52.35 

30 

6.11 

12.22 

18.32 

24.4-3 

30.54 

36.65!   42.75     48.86 

54.97 

61.08 

4 

6.98 

13.90 

20.91 

27.93     34.90 

41.88     48.86     55.84 

06.82 

69.80' 

30 

7.85 

15.70 

33.5'j 

31.41     39.26 

47.11 

54.96 

62.82 

70.67 

78.52 

5 

8.73 

17.45 

26.17 

34.89 

43.62 

52.134 

61.07     69.79 

78.51 

87.24 

30 

9.60 

19.19    28.79 

38.33 

47.98 

57.57 

67.17 

76.70 

86.36 

95.96 

6 

10.47 

20.93    31.40, 

41.87 

53.34 

62.80 

73.27     83.74 

94.20 

104.67 

30 

11.34 

22.68    34.03 

45.35 

56.67 

68.03 

79.37     90.71 

102.05 

113.39 

7 

13.21 

24.42    36.63 

48.84 

61.05 

73.26 

85.47 

97.08 

109.89 

122.10 

30 

13.03 

26.16    39.24; 

52.32 

65.40 

78.48 

91.56  104.64 

117.73 

130.81 

g 

13.95 

27.901  41.85! 

55.80 

69.76 

83.71 

97.00   111.01 

125.56 

139.51 

30 

14.82 

29.64i  44.47; 

59.29 

74.11 

88.93 

103.75 

118.57 

1&3.40 

148.22 

9 

15.69 

31.38:  47.03! 

62.77 

78.46 

94.15 

109.84 

125.53 

141.23 

156.92 

30 

16.56 

33.12 

49.68j 

66.25 

82.81 

99.37 

115.93 

182.49 

149.05 

165.62 

10 

17.43    34.86 

52.29 

69.72 

87.16 

104.59 

122.02 

139.45 

150.88)174.31 

30 

18.30    36.60 

54.90 

73.20 

91.50 

109.80 

128.10 

146.40 

164.701183.00 

11 

19.17 

38.34 

57.51 

76.68 

95.85 

115.01 

134.18 

153.  So 

172.52 

191.69 

30 

20.04 

40.03 

60.11 

80.15 

100.19 

120.23 

140.26 

160.30 

180.34 

200.38 

12 

20.91 

41.81 

62.72 

83.62 

104.53 

125.43 

140.34 

107.25 

188.15 

209.06 

30 

21.77 

43.55 

65.32 

87.09 

108.87 

130.1  54 

152.41 

174.19 

195.96 

217.73 

13 

23.64 

45.23 

67.92 

90.56 

113.20 

135.84 

158.48 

181.13 

203.77 

226.41 

30 

23.51 

47.01 

70.521 

94.03 

117.54 

141.04 

164.55 

188.00 

211.57 

235.07 

14 

24.37 

48.75 

73.12 

97.50 

121.87 

140.24 

170.62 

194.99 

219.36 

243.74 

30 

25.24 

50.48 

75,72 

100.90 

126.20 

151.44 

170.68 

201.92 

227.16 

252.40 

15 

26.11 

52.21 

78.321  104.42 

130.53 

156.  63 

182.74 

208.84 

234.95 

261.05 

30 

2<5.97 

5'3.!»  4    80.91 

107.83 

1:34.  85 

161.82 

188.79 

215.70 

242.73 

269  70 

16 

37.83 

55.67:  83.50 

111.34 

139.17 

167.01    194.84 

222.08 

250.51 

278  .'35 

80 

2170 

57.40    88.10 

114.79 

143.49 

17'2.19   200.89 

229.59 

258.29 

286.99 

17 

2;).  56 

59.12;  83.69 

118.23 

147.81 

177.37 

'•'00  ').') 

236.50 

266.06 

295.62 

30 

30.43 

00.85    91.27; 

121.70 

153.12 

182.55 

212^97 

243.40 

273.82 

304.25 

18 

31.23 

03.57    93.33 

125.15 

156.43 

187.72 

219.01 

250.30 

281.58 

312.87 

80 

32.15 

04.3)    93.45 

123.59 

160.74 

192.89 

225.04 

257.19 

289.34 

321.49 

19 

33.01 

00.02    90.03 

132.04 

105.05 

198.06 

231.07 

204.08 

297.08 

330.09 

30 

33.87 

67.7,  131.01 

135.43 

169.35 

203.22 

237.09 

270.96 

304.83 

338.70 

20 

34.73 

69.46  104.19  138.92 

173.65 

208.38 

243.11 

277.84 

312.57 

347.30 

30 

.35.59 

71.13  108.77! 

142.35 

177.94 

213.53 

249.12 

284.71 

320.30 

&55..S9 

21 

36.45 

73.89  109.34 

145.79 

183.24 

218.68 

255.13 

291.58 

328.02 

364.47 

30 

37.30 

74.61  111.91 

149.22 

186.52  223.83 

261.13 

298.44 

335.74 

373.05 

22 

33.16 

76.33  114.49 

152.65 

190.81   228.97 

267.13 

305.20 

343.46 

381.62 

30 

39.02 

78.04  117.05 

156.07 

195.00  234.11 

27'3.13 

312.14 

&51.16 

390.18 

23 

39.87 

79.75  119.63 

159.49 

199.  37  i  239.  24 

279.12 

318.89 

S58.86 

398.74 

30 

40.73 

81.46  133.19 

168.91 

203.641244.37 

285.10 

325.83 

866.50 

407.28 

21 

41.53 

83.16  134.75 

1:56.33 

207.  911849.  49 

291.08 

332.06 

37'4.24 

415.82 

30 

43.44 

84.87  137.31: 

109.74 

212.18:254.61 

297.05 

330.48 

381.92 

424,36 

25 

43.29! 

86.58!129.8G! 

173.15 

216.44:259.73 

303.02 

364.30 

389.59 

432.88 

30 

44.  14' 

83.  33  :  133.  43 

176.50 

220.70  264.84 

308.98 

353.12 

397.26 

441.39 

26 

44.99 

89.98  134.97 

179.9(5 

234.95  269.94 

314.913 

859.92 

404.91 

449,90 

30 

45.84 

91.68  137.53 

183.30 

229.20.275.04 

330.88 

366.72 

412.56 

458.40 

27 

46.69 

93.38  140.07 

180.70 

233.45  280.14 

33(5.83 

373.51 

420.20 

466.89 

30 

47.54 

95.07  142.61 

190.15 

237.09  285.22 

332.7(5 

380.30 

427.83 

475.37 

28 

43.33 

96.77  145.15 

193.5! 

241.93  290.31 

3:38.09 

387.08 

4:35.46 

4R3.84 

30 

49.23) 

98.46  147.09 

196.92 

246.15  295.38 

344.63 

393.85 

443.08 

492.31 

29 

50.08 

100.15  150.23 

200.30 

250.38  300.46 

350.53 

400.10 

450.68 

500.76 

30 

50.92 

101.84  152.76 

203.08 

254.00  305.52 

350.44 

407.36 

458.28 

509.20 

30 

51.76 

103.53  155.  291207.  06 

258.  82!  310.  59 

362.35 

414.11  465.  87  1517.  64 

301 


TABLE  X. -COEFFICIENTS  FOR  VALVOID  ARCS. 


I.—  RATIO  OF 

u  =  - 

t 

A 

L 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

90° 

100° 

110° 

120° 

300 

.3518J 

.3516 

.3514 

.3510 

.3506  .a500 

.3493 

.3485 

.3476 

.3466 

.S455 

.3444 

400 

.3437 

.3436 

.3433 

.3430 

.3426 

.3421 

.3415 

.3408 

.3399 

.3390 

.3:380 

.3368 

500 

.3400 

.3398 

.3396 

.3393 

.3389 

.3383. 

.3379 

.3373 

.3364 

.3356 

.3345 

.3835 

600 

.3379 

.3378 

.3376 

.3373 

.3369 

.3365 

.3359 

.3353 

.3:345 

3337 

.3327 

.8317 

700 

.3367 

.3366 

.3364 

.3361 

.aS57 

.3:353 

.3347 

.aS41 

.3334 

'.332(j 

.:3316 

.3306 

800 

.3359 

.3358 

.3356 

.3a53 

.3349 

.3345 

.3340 

.3333 

.3326 

asis 

.3309 

.329;) 

900 

.3353 

.3352 

.3350 

.3344 

.3340 

.3334 

.3328 

.!3321 

'.3313 

.330-1 

.3294 

1000 

.3350 

.3348 

.3346 

'.3344 

.3340 

.aase 

.3331 

.3324 

.3317 

.3310 

.3301 

.3291 

1200 

.3345 

.a343 

.3341 

.3339 

.3336 

.  3331 

.3326 

.3320 

.13313 

.3305 

.3296 

.3286 

1500 

.3340 

.8339 

.aS37 

.3335 

.3331 

.£327 

.3322 

.asi6 

.3309 

.3301 

.3292 

.3283 

2000 

.3337 

.3336 

.3333 

.3331 

.3328 

.3324 

.3319 

.3313 

.3306 

.3298 

.3289 

.3280 

n.—  RATIO  OF 

r 

v  =  —  • 

L, 

L 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

90° 

100° 

110° 

120° 

300      .7706 

.7683 

.76431.7588 

.75181.7432 

.7aS2 

.7218  .7090  .6949 

.6795 

.6630 

400      .7611 

.7588 

.7549  .7495 

.7425  .73-11 

.  7243 

.7130 

.7004 

.6865 

.6714 

.6551 

500 

.7568 

.7'545 

.7506  .7452 

.7384  .7300 

.7202 

.7091 

.69(56  .6828 

.6678 

.6516 

600 

.7545 

.7522 

.7483  .7430 

.73611.7278 

.7181 

.7070 

.6946 

.6808 

.6659 

.6498 

700 

.7531 

.7508 

.7469 

.7416 

.73481.7265 

.7168 

.7057 

.69:33  .6797 

.6648 

.6487 

800 

.7522 

.7499 

.7461 

.7407 

.73391.7257 

.7160 

.7040 

.6926 

.6789 

.6640 

.6480 

900 

.7516 

.7492 

.7454 

.7401 

.7888.7251 

.7151 

.7041  .6920  .678-1 

.6635 

.6475 

1000 

.7512 

.7489 

.7450 

.7397 

.7329  .7247 

.7150 

.704< 

.(5917 

.6780 

.6682 

.6472 

1200 

.7505 

.7483 

.7444 

.7391 

.7324  .7241 

.7145  .7035   .6912  .67751.6627     .6468 

1500 

.7501 

.7478 

.7440  .7887 

.73191.7237 

i.7141 

.7031 

|.69Cte 

.6772 

.(5624 

.6464 

2000 

.7497 

.7474 

.7436 

.7383 

.7316 

.7234 

i.7137 

.702* 

(  .69041.6769 

.6621 

.6461 

JTT  RATIO 

I 

' 

A'  -  A" 

TO  A  CHANGE  OF  ONE  DEGREE  IN  THE  ANGLE  A. 

L 

10° 

20°   i  30° 

40° 

50° 

60° 

70° 

80° 

90° 

100° 

110° 

120° 

300 

2.62 

2.61  1  2.60    2.59 

2.57 

2.55 

2.52 

2.49 

2.46 

!    2.42 

2.38 

2.34 

400 

3.49 

3.48 

3.46!  8.44 

3.42 

3.38 

3.35 

3.30 

i  3.25 

I    3.20 

3.14 

3.08 

500 

4.36 

4.35 

4.33J  4.30 

4.26 

4.22    4.17 

4.11 

4.05i    3.98 

3.90 

3.81 

600 

5.23 

5.22 

5.19    5.10 

5.11 

5.06 

1  4.99 

4.92 

4.84 

4.75 

4.65 

i     4.55 

700 

6.10    6.09 

6.06    6.02 

5.96 

5.90 

i  5.83 

5.74 

5.651   5.54 

5.43 

5.31 

800 

6.97 

6.95 

6.92!  6.87 

6.82 

6.74 

i  6.66 

6.56 

6.45 

6.  as 

6.20 

6.06 

900 

7.85 

7.79    7.73 

7.67 

7.59 

1  7.49 

7.38 

!  7.26 

7.13 

6.98 

;     6.82 

1000 

8.72 

8  69 

8.65i  8.59 

8.52 

8.43 

8.32 

8.20 

8.07 

i    7.92 

7.75 

7.58 

1100 

9.59 

9^56 

9.52    9.45 

9.37|  9.27 

!  9.16 

i  9.02 

8.87 

i    8.71 

8.53 

8.34 

120) 

10.46 

10.43 

10.38  10.31 

10.22  10.11 

!  9.99 

!  9.84 

9.68 

9.50 

9.31 

9.09 

1300 

11.33 

11.30 

11.25  11.17 

11.07  10.96 

10.8'i 

10.66 

10.49 

S10.29 

10.08 

9.85 

1400 

12.21 

12.17 

12.  11  12.  as 

11.93  11.80 

ill.  65 

11.48  11.29 

111.  08110.  86 

10.61 

1500 

13.08 

13.04 

12.98!12.89 

12.78!l2.64 

12.48 

12.30 

12  10 

11.88 

11.68 

!  11.37 

1600 

13.95 

13.91 

13.84  13.75.13.63 

13.49 

13.32 

13.12  12.91 

12.67 

12.41 

12.13 

1700 

14.82 

14.78 

14.71  14.61 

14.48 

14.33 

14.15 

13.94 

13.71 

13.46 

13.18 

12.88 

1800 

15.69 

15.65 

15.5715.47 

15.33 

15.17 

14.98 

14.76  14.52  14.25 

13.96 

13.64 

1900 

16.57 

16.52 

16.44116.33 

16.19116.  01 

15.81 

15.58 

is.  as 

15.04 

14.73 

i  14.40 

2000 

17.44 

17.39 

17.30il7.19 

117.04  16.86 

16.  65  16.  40  1613  15.83 

15.51 

15.16 

302 


TABLE  XL-TURNOUTS    AND    SWITCHES    FROM    A    STRAIGHT 
TRACK.     §§180,  181,  182. 


GAUGE,  4  FEET  8^3  INCHES  =  4 

.708.    THRO  w,  5  INCHES  =  0  .  41  7. 

No. 

Angle 

Dist. 

Chord  i  Switch      Radius      Log'thm. 

Degree 

n. 

F. 

BF. 

Of. 

AD. 

r. 

log.  r. 

of  Curve. 

4         14°  15'  00" 

37.664       37.373 

11.209         150.656 

2.177986 

38°  45'  57" 

4)^  j  12 

10  49 

42 

.372 

4 

2.113 

1 

2.610  i       190. 

674 

2.2S 

«292 

I 

)  24  09 

5         11   25  16 

47.080 

46.846 

14.012        235.400 

2.371806 

24  31  36 

10 

23  20 

51 

.188 

5 

1.575 

1 

5.413         284 

834 

2Ai 

>4592 

2( 

)  13  13 

6  ~ 

9  31   39 

56.496 

56.301 

16.814         338 

1)76 

2.530169 

]( 

J  57  52 

OK 

8  < 

17  51 

61 

.204 

6 

1.024 

1 

8.215         397 

820 

2.5t 

mm 

ft 

I  26  25 

7 

8  10  16 

65.912 

65.744 

19.616  i       461 

384 

2.664063 

12  26  34 

T^ij 

7  « 

57  41 

70 

.620 

r* 

0.464 

2 

1.017 

529 

(i.-,0 

2.7$ 

S989 

1( 

)  50  02 

8  " 

7  ( 

)9  10 

It 

.328 

7 

5.181 

2 

2.418 

602 

2.7S 

S0046 

! 

)  31  07 

8V£ 

6  43  59 

8 

.036 

79.898 

23.820 

680 

3<V> 

2.832704 

I 

J  25  47 

9 

6  5 

11   35 

81 

.744 

8 

4.613 

2 

5.221 

762 

(Mi 

2.« 

2352 

r  31  04 

6  01   32 

H 

.452 

89.328 

26.622 

849 

704 

2.929314 

i 

J  44  46 

10  ~ 

5  <• 

13  29 

94 

.160 

9 

4.043 

2 

8.023 

941 

ooo 

2.9r 

'3866 

( 

5  05  16 

5  27  09 

98.868 

98.756 

i 

9.424 

1038 

114 

3.01 

6245 

5  31  17 

11  S 

5  1 

2  18 

103 

.576 

10 

3.469 

.  a 

0.825 

1139 

330 

3.  Of 

.6652 

>  01   50 

HJ^ 

4  i 

rfj  45 

108 

.284 

10 

S.182 

3 

2.227 

1245. 

21  M; 

3.0J 

5262 

^ 

I  36  08 

12 

4  4 

6   19 

112.992 

112.894 

33.628 

1355. 

!HM 

3.132229 

4  13  36 

GAUGE,  3  FEET.    THROW,  4  INCHES  =  0.333. 

No. 

Angle 

Dist. 

Chord  1  Switch     Radius 

Log'thm. 

Degree 

n. 

F. 

BF. 

a/. 

AD. 

r. 

log. 

r. 

of  Curve. 

4 

14° 

15'  00" 

24 

23.815 

8              96.0 

1.982 

•271 

6$ 

1°  46'  34" 

4/^ 

12 

40  49 

27 

26.835 

9 

121. 

5 

2.084576 

48  36  04 

5 

11 

25  16 

30 

2 

9.851 

10 

150. 

0 

2.176 

091 

i 

!  56  33 

5^ 

10 

23  20 

as 

3 

2.865 

11 

181. 

5 

2.25S 

#77 

31 

58  55 

6 

9 

31   39 

36 

35.876 

12 

216.0 

2.334454 

26  46  07 

6V> 

8 

47  51 

39 

3 

8.885 

13 

253. 

5 

2.40S 

978 

2$ 

J  45  04 

7  " 

8 

10   16 

42 

41.893 

14 

294. 

0 

2.46£ 

,347 

19  35  01 

7V> 

7 

37  41 

45 

4 

4.900 

15 

v  337. 

5 

2.52* 

274 

r 

'  02  21 

8  ~ 

7 

09  10 

48 

4 

7.906 

16 

384. 

0 

2.584 

331 

14 

I  57  48 

8^> 

G 

43  59 

51 

50.912 

17 

433. 

5 

2.636989 

13  14  47 

9  ~ 

G 

21   35 

54 

5 

3.917 

18 

486. 

0 

2.68(3 

636 

11 

48  37 

i)V« 

G 

01  32 

57 

5 

6.921 

19 

541. 

5 

2.73S 

598 

H 

)  35  46 

10  ~ 

5 

43  29 

60 

59.925 

20 

600. 

0 

2.778151 

< 

>  33  38 

lOVo 

5 

27  09 

63 

6 

2.929 

21 

661. 

5 

2.82C 

530 

i 

i  40  12 

11  ~ 

5 

12  18 

66 

65.932 

22 

726. 

0 

2.86C 

1937 

7  53  54 

11*6 

4 

5S  45 

69 

6 

8.935 

23 

793. 

5 

2.89S 

547 

r 

*  13  32 

12 

4 

46  19 

72 

71.938 

24 

864. 

0 

2.936514 

( 

>  £8  06 

ANGLE  AND 

DISTANCE  OF  MIDDLE 

FROG,  F' 

Gauge 

Gauge 

Gauge 

Gauge 

No.     No. 

Angle 

4,  8Vjj>. 

3. 

No. 

No. 

An^le 

4,  8Va- 

3. 

w.       n". 

F". 

Dist. 

Dist.  i 

n. 

n*. 

F* 

Dist 

Dist. 

aF'. 

of:  I 

aF". 

aF". 

4       2.817 

20°  07'  36" 

26.736 

17.037 

8 

5.651 

10°  06'  44" 

53.317 

33.974 

4U    3.172 

17  54  52 

30.054 

19.151 

6.005 

9  31  08 

56.643 

36.094 

5       3.527 

16  08  19 

33  374 

21.266  i 

9 

6.359 

8  59  30 

59.969 

38.213 

5Va   3.881 

14  40  58 

36.695 

23.383 

6.713 

8  31   10 

63.296 

40.333 

6       4.235 

13  27  57 

40.018 

25.500 

10 

7.067 

8  05  40 

66.623 

42.458 

6V«    4.589 

12  26  07  |  43.342 

27.618 

7/420 

7  42  35 

69.950 

44.573 

7       4.943 

11   33  04 

46.666 

29.736 

11 

7.774 

7  21   36 

73.277 

46.693 

7U    5.297 

10  47  02 

49.991 

31.855 

8.128 

7  02  26 

76.605 

48.813 

8       5.651   10  06  44 

53.317 

33.974 

12       8.482 

6  44  51 

79.932 

50.934 

803 


TABLE  XII.— MIDDLE  DEDICATES  FOR  CURVING*  RAILS.     §103. 


LENGTH  OF 

RAIL-CHORD. 

~  1 

D 

D 

32 

30 

28 

26 

24 

22 

s 

10 

18 

16       14 

12 

10 

1° 

.022 

.020 

.017 

.015 

013 

.011 

".009 

.007 

.006 

.004 

.003 

.002 

1° 

2 

.045 

.039 

.034 

.030 

025 

• 

0^1 

.( 

m 

.01 

4 

.011 

.009 

.0 

ti 

.004 

2 

3 

.067 

.059 

.051 

.044 

!038  '   .032     .026 

.021 

.017    .013 

.009 

.007 

3 

4 

.089 

.079 

.0138 

.059 

050 

:, 

04^ 

«5 

.Oi 

S 

.022    .017 

.0 

u 

009 

4 

5 

.112 

.098 

.036 

.074 

Oc3 

053 

.'044 

;085 

.028    .021 

.016 

.011 

5 

6 

.134 

.118 

.103 

.088 

075 

^ 

U03 

tt2 

.OJ 

-.' 

.034    .026 

.013 

ft 

7 

.156 

.137 

.120 

.103 

.'(H8 

on 

'.061 

.049    .039 

.0:30 

.022 

.015      7 

8 

.179 

.157 

.137 

.118 

100 

a 

w 

.( 

KTO 

.0£ 

7 

.045 

.0:34 

.0 

25 

.017!    8 

9 

.201 

.177 

.154 

.133 

J13 

Hto 

.078 

.064 

.050    .0:38 

.028 

.020      9 

10 

.223 

.196 

.171 

.147 

126 

105 

)S7 

.07 

1 

.05(5    018 

.0 

81 

.022  '  10 

11 

.245 

.216 

.188 

.162 

138 

Hi} 

!096 

.078 

.061 

.047 

.035 

.024;  11 

12 

.268 

.235 

.205 

.177 

151 

127 

05 

.0!- 

15 

.067 

.051 

.0 

:;s 

.026    12 

14 

.312 

.274 

.2:38 

.206 

175 

147 

!l22 

.099 

.078 

.<60 

.014 

.030    14 

16 

.356 

.313 

.273 

.2:35 

200 

KiS 

i 

39 

.11 

8 

.089 

.068 

.0 

50 

.035    16 

18 

.400 

.352 

.307 

.264 

225 

189 

'l56 

.127 

.100 

.077 

.056 

.039    18 

20 

.445 

.391 

.340 

.293 

250 

j 

MO 

71 

.14 

1 

.111 

.085 

M 

.043    20 

24 

.531 

.467 

.40?' 

.351 

299 

f 

r.i 

~> 

J07 

.!( 

s 

.133 

.102 

.0 

75 

.052    24 

28 

.618 

.543 

.473 

.408 

347 

'.241 

.195 

.154 

.118 

.087 

.060    28 

33 

.705 

.619 

.539 

.465 

396 

( 

V» 

i 

275 

.!& 

8 

.176 

.135 

.0 

99 

.069    32 

36 

.791 

.696 

.603 

.522 

445 

ft';} 

i 

309 

.250 

.197 

.151 

.111 

.077    36 

40 

.878 

.772 

.672 

.579 

493 

114 

., 

U2 

.2" 

7 

.219, 

.168 

.1 

88 

.086    40 

45 

.983 

.863 

.752 

.648 

552 

!463 

.383 

.305 

.188 

.137 

.096  i  45 

50  jl.087 

.955 

.831 

.716 

610 

.512 

.423 

.343 

^271 

.207 

.152 

.106 

50 

TABLE  XIII.  -DIFFERENCE 

IN 

ELEVATION   OF   RAILS  ON 

CURVES 

§201. 

VELOCITY  IN 

MILES  PER  HOUR. 

D 

D 

10         15        20        25 

30 

35 

40 

45 

50 

60 

1 

.006 

.013      .023      .03 

i 

.051 

.070 

191 

.116  !.    .143 

206       1 

2 

.Qll 

.026      .046      .071 

.103 

.140 

188 

.231 

.285 

410  1    2 

3 

.017 

.039      .069      .10 

.If 

4 

.2 

10 

874 

.346 

.42 

7 

612       3 

4 

.023      .051      .091      .143 

.20 

6 

.9 

<0 

!365 

.461 

.568 

.811       4 

5 

.029 

.064      .114      .17 

1 

.2 

7 

.a 

J',» 

455 

.574 

.7(1 

7 

1 

006 

5 

6  1   .0:34 

.077      .137      .214 

.3C 

18 

.418 

.'545 

.687 

.844 

1 

196 

6 

7      .040 

.090      .160      .25 

) 

.3£ 

g 

.41 

37 

( 

ttl 

.798 

.97 

g 

8      .046 

.103      .183      .285 

.410 

.556 

i'23 

.908 

1.112 

9      .(151 

.116      .206      .32 

) 

.4fi 

«i 

.6, 

.'l 

•>]  1 

1 

.017 

10      .057 

.129      .228      .35 

1 

.51 

i 

.(i 

12 

^'.)(v>. 

1 

.124 

11 

.063 

.142      .251      .391 

.561 

7 

10 

!984 

12 

.069 

.154      .274      .42 

.61 

t 

.» 

6 

1. 

JO!) 

14  !   .080 

.180      .319       497 

.711 

,!» 

16 

.091      .206      .365       56 

.80 

!) 

1  Ot 

>8 

18      .102      .231      .410      .637 

.906 

20      .114      .256      .4oo      .70 

1.00 

2 

25      .141      .318      .563      .77 

5 

30      .168 

.380      .672      .844 

35 

.195      .441      .778 

40 

.222      .501      .831 

50 

.276  1   .618 

80-1 


TABLE  IIY.-GRADES  AM)  GRADE  ANGLES. 


TABLE  XIV.— GRADES  AND  GRADE  ANGLES. 


Feet 
per 
Sta- 
tion. 

Feet  per 
Mile. 

Inclina- 
tion. 

Feet 
per 
Sta- 
tion. 

Feet  per 
Mile. 

Inclina- 
tion. 

Feet 

S£ 

tion. 

Feet  per 
Mile. 

Inclin- 
ation. 

0       /        • 

0        /        » 

• 

0        /         • 

.01 

.528 

21 

.51 

26.928 

17  32 

1.01 

53.328 

34  43 

.02 

1.056 

41 

.52 

27.456 

17  53 

.02 

53.856 

35  04 

.03 

1.584 

1  02 

.58 

27.984 

18  13 

.03 

54.384 

35  24 

.04 
.05 

2.112 
2.640 

1  23 

1  43 

.54 
.55 

28.512 
29.040 

18  34 
1854 

.04 

.05 

54.912 
55.440 

35  45 
36  05 

.06 

3.168 

2  04 

.56 

29.568 

19  15 

i     .06 

55.968 

36  26 

.07 

3.696 

2  24 

.57 

30.096 

19  36 

!     .07 

56.496 

36  47 

.08 

4.224 

245 

.58 

30.624 

19  56 

.08 

57.024 

37  08 

.09 

4.752 

3  06 

.59 

31.152 

20  17 

!     .09 

57.552 

3728 

.10 

5.280 

3  26 

.60 

31.680 

20  38 

;  l.io 

58.080 

3749 

.11 

5.808 

3  47 

.61 

32.208 

20  58 

.11 

58.608 

38  09 

.12 

6.336 

4  08 

.62 

32.736 

21  19 

!     .12 

59.136 

38  30 

.13 

6.864 

4  28 

.63 

33.264 

21  39 

!     .13 

59.064 

38  51 

.14 

7.392 

4  49 

.64 

33.792 

22  00 

1     -14 

60.192 

39  11 

.15 

7.920 

509 

,65 

34.320 

22  21 

.15 

60.720 

39  32 

.16 

8.448 

5  30 

.66 

34.848 

2241 

!     .16 

C1.248 

39  53 

.17 

8.976 

5  51 

.67 

35.376 

23  02 

.17 

C1.776 

40  13 

.18 

9.504 

6  11 

.68 

35.904 

23  23 

.18 

62.304 

40  34 

.19 

10.032 

6  32 

.69 

36.432 

23  43 

.19 

02.832 

40  54 

.20 

10.560 

6  53 

.70 

36.960 

2404 

.20 

63.360 

41  15 

.2T 

11.088 

7  13 

.71 

37.488 

24  24 

.21 

63.888 

41  35 

.22 

11.616 

7  34 

.72 

38.016 

24  45 

.22 

64.416 

41  56 

.23 

12.144 

7  54 

.73 

38.544 

25  06 

.23 

64.944 

42  17 

.24 

12.672 

8  15 

.74 

39.072 

25  26 

.24 

65.472 

42  38 

.25 

13.200 

8  36 

.75 

39.600 

25  47 

.25 

66.000 

42  58 

.26 

13.728 

8  56 

.76 

40.128 

2608 

.26 

C6.528 

43  19 

.27 

14.256 

9  17 

.77 

40.656 

26  28 

.27 

67.056 

43  39 

.28 

14.784 

938 

.78 

41.184 

2649 

.28 

07.584 

44  00 

.29 

15.312 

9  58 

.79 

41.712 

27  09 

.29 

68.112 

44  21 

.30 

15.840 

10  19 

.80 

42.240 

27  30 

.30 

68.1540 

44  41 

•  .31 

16.368 

1039 

.81 

42.768 

27  51 

.31 

69.168 

45  02 

.32 

16.896 

11  00 

.82 

43.296 

28  11 

.32 

69.696 

45  23 

.33 

17.424 

11  21 

.83 

43.824 

28  32 

..33 

70.224 

45  43 

.34 

17.952 

11  41 

.84 

44.352 

28  53 

.34 

70.752 

4604 

.35 

18.480 

12  02 

.85 

44.880 

29  13 

.35 

71.280 

46  24 

.36 

19.008 

12  23 

.86 

45.408 

2934 

.36 

71.808 

4645 

.37 

19.536 

1243 

.87 

45.936 

29  54 

.37 

72.336 

47  06 

.38 

20.064 

13  04 

.88 

46.464 

30  15 

.38 

72.864 

47  26 

.39 

20.592 

1324 

.89 

46.992 

30  36 

.39 

73.392 

47  47 

.40 

21.120 

13  45 

.90 

47.520 

30  57 

.40 

73.920 

4808 

.41 

21.648 

14  06 

.91 

48.048 

31  17 

.41 

74.448 

4828 

.42 

22.176 

14  26 

.92 

48.576 

31  38 

.42 

74.976 

48  49 

.43 

22.704 

14  47 

.93 

49.104 

31  58 

.43 

75.504 

49  09 

.44 

23.232 

15  08 

.94 

49.632 

32  19    i 

.44 

76.032 

49  30 

.45 

23.760 

15  28 

.95 

50.160 

32  39 

.45 

76.560 

49  51 

.46 

24.288 

15  49 

.96 

50.688 

33  00 

.46 

77.088 

50  11 

.47 

24.816 

1609 

.97 

51.216 

33  21 

1.47 

77.616 

50  32 

.48 

25.344 

16  30 

.98 

51.744 

33  41 

1.48 

78.144 

5052 

.49 

25.872 

16  51 

.9!) 

52.272 

3402 

1.49 

78.672 

51  13 

.50 

26.400 

1711 

1.00 

52.800 

3423 

1.50 

79.200 

51  34 

C05 


TABLE  XI\r.—  GRADES 


GRADE  ANGLES. 


Feet 
per  j 
Sta- 
tion. 

Feet  per 
Mile. 

Inclina- 
tion. 

Feet 
per 
Sta- 
tion. 

Feet  per 
Mile. 

|| 

Inclina- 
tion. 

Feet 
per 
Sta- 
tion. 

Feet  per 
Mile. 

Inclina- 
tion. 

at" 

0        /        » 

0         /         » 

51 

79.728 

51  54 

2.05 

108.240 

1  10  28 

5.10 

269.280 

2  55  10 

52 

80.256 

52  15 

2.10 

110.880  )  1  12  11 

5.20 

874.660 

25836 

.53       80.784 

52  36 

2.15 

113.520      1  13  54 

5.30 

219.840 

3  02  09 

54       81.312 

52  56 

2.20 

116.160      1  15  37 

5.40 

285.120 

3  05  27 

.55       81.840 

53  17 

2.25 

118.800  j  1  17  20 

5.50 

290.400 

3  08  53 

.56       82.368 

53  37 

2.30 

121.440      1  19  03 

5.60 

295.680 

3  12  19 

.57       82.896 

53  58 

2.35 

124.080      1  20  46 

5.10 

300.  S60 

3  15  44 

.58  |     83.424 

54  19 

2.40 

126.720      1  22  29 

5.80 

806.240 

3  19  10 

.59       83.952 

54.39 

2.45 

129.360      1  24  12 

5.90 

311.520 

3  22  36 

.60 

84.480 

55  00 

2.50 

132.000 

1  25  56 

6.00 

316.800 

326  01 

.61 

85.008 

55  21 

2.55 

134.640 

1  27  89 

6.10 

822.080 

3  29  27 

.62 

85-536 

55  41 

2.60 

137.280 

1  29  22    I  6.20 

827.860 

3  32  52 

.63       86.064 

56  02 

2.65 

139.920 

1  31  05    |  6.10 

882.640 

3  36  18 

.64       86.592 

56  22 

2.70 

142.  £60 

1  32  48 

6.40 

837.920  1  3  89  43 

.65       87.120 

56  43 

2.75 

145.200 

1  34  31 

6.  CO 

343.  2CO 

3  43  08 

.66  !     87-648 

57  04 

2.80 

147.840 

1  86  14 

6.  tO 

348.480 

3  46  34 

.67       88.176 

57  24 

2.85 

150.460 

1  37  57 

6.10 

353.  7CO 

349  59 

.68 

88.704 

57  45 

2.90" 

153.120 

1  39  40 

6.80 

359.040 

3  53  24 

.69 

89.232 

58  06 

2.95 

155.760 

1  41  23 

6.£0 

864.820 

3  56  50 

.70  i     89.760 

58  26 

3.00 

158.400 

1  43  06 

7.CO 

869.600 

4  CO  15 

.71       90.288 

58  47 

3.05 

161.040 

1  44  49 

7.10 

S74.880 

4  03  40 

.72  i     90.816 

59  07 

3.10 

163.680 

1  46  82 

7.20 

880.160 

4  0706 

.73  !     91.344 

59  28 

3.15 

166.320 

1  48  15 

7.  SO 

885.440 

4  10  31 

.74 

91.872 

59  49 

3.20 

168.960 

1  49  58 

7.40 

SCO.  120 

4  13  56 

.75 

92.400 

1  00  09  , 

3.25 

171.600 

1  51  41 

7.50 

SC6.0CO 

4  17  21 

.76 

92.928 

1  00  30  ' 

3.30 

174.240 

1  53  24 

7.60 

401.280 

4  20  46 

.77 

93.456 

1  00  51  I 

3.35 

176.880 

1  55  07 

7.70 

4C6.5CO 

4  24  11 

.78 

93.984 

1  01  11 

3.40 

179.520 

1  £6  50 

7.80 

411.840 

4  27  86 

.79 

94.512 

1  01  32 

3.45 

182.160 

1  £8  83 

7.  GO 

417.120 

4  81  01 

.80 

95.040 

1  01  52 

3.50 

184.800 

2  CO  16 

8.  CO 

422.  4CO 

4  84  26 

.81 

95.568 

1  02  13 

3.55 

187.440 

2  01  59 

8.10 

427.680 

4  87  51 

.82 

96.096      1  02  34  i 

3.60 

190.080     2  03  42 

8.20 

482.  9CO 

4  41  16 

.83 

96.624 

1  03  64 

3.65 

192.720      2  05  25 

8.80 

488.240 

4  44  41 

.84 

97.152 

1  03  15 

3.70 

195.360     2  07  08 

8.40 

443.520 

4  48  C6 

.85 

97.680 

1  0335 

3.75 

198.000     2  08  51 

8.EO 

448.  8CO 

4  51  80 

.86  i     98.208 

1  03  56 

3.80 

200.640 

2  10  34 

8.60 

454.  C80 

4  54  55 

.87       98.736 

1  04  17 

3.85 

203.280 

2  12  17 

8.70 

459.860 

4  58  20 

.88 

99.264 

1  04  37 

3.90       205.920 

2  14  00 

8.80  1     464.640 

5  01  44 

.89 

99.792 

1  04  58 

3.95       208.560 

2  15  43 

8.90 

469.920 

5  05  10 

.90 

100.320 

1  05  19 

4.00 

211.200 

2  17  26 

9.00 

415.  2CO 

5  C8  34 

.91 

100.848 

1  05  39 

4.10 

216.480 

220  52 

9.10 

480.480 

5  11  59 

.92 

101.376 

1  06  00 

4.20 

221.760 

2  24  18 

9.20 

485.  7CO 

5  15  23 

.93 

101.904 

1  06  20 

4.30 

227.040 

2  27  44 

9  30 

491.040 

5  18  48 

.94 

102.432 

1  06  41 

4.40 

232.320 

2  31  10 

9.40 

496.320 

5  22  12 

.95 

102.960 

1  07  02 

4.50 

237.600 

2  34  36 

9  50 

501.  6CO 

5  25  37 

.96 

103.488 

1  07  22 

4.60 

242.880 

23801 

9.60 

506.880 

5  29  01 

.97 

104.016 

1  07  43 

4.70 

248.160 

241  27 

9.70 

512.  ICO 

5  32  25 

1.98 

104.544 

1  08  04 

4.80 

253.440 

2  44  53 

9.80 

517.440 

5  35  50 

1.99 

105.072 

1  0824 

4.90 

258.720 

2  48  19 

9.90 

522.720 

5  39  14 

2.00 

105.600 

1  08  45 

5.00 

264.000 

2  51  45 

10.00 

528.000 

5  42  38 

306 


TABLE  XV.— FOR  OBTAINING  BAROMETRIC  HEIGHTS  IN  FEET. 


Barom- 
eter. 
Inches 

0.00 

0.02 

0.04 

0.06 

0.08 

Diff.  per 

.002  in. 

19°.  0 
.1 
.2 

16832 
16970 
17107 

16860 
16997 
17134 

16888 
17025 
17162 

16915 
17052 

17189 

16943 
17080 
17216 

2.8 
2.8 
2.7 

.3 

17243 

17270 

17298 

17325 

17852 

2.7 

.4 

17379 

17406 

17433 

17460 

17487 

2.7 

.5 

17514 

17540 

17567 

17594 

17621 

2.7 

.6 

17648 

17674 

17701 

17728 

17755 

2.7 

.7 

17781 

17803 

17&34 

17861 

17887 

2.7 

.8 

17914 

17940 

17967 

17993 

18020 

2.7 

.9 

18046 

18072 

18099 

18125 

18151 

2.6 

20°.  0 

18178 

18204 

18230 

18256 

18282 

2.6 

.1 

18308 

18334 

18360 

18386 

18413 

2.6 

.2 

18438 

18464 

18490 

18516 

18542 

2.6 

18568 

18594 

18820 

18645 

18671 

2.6 

A 

18697 

18723 

18748 

18774 

18799 

2.6 

.5 

18825 

18851 

18S76 

18902 

18927 

2.6 

.6 

18953 

18978 

19004 

19029 

19054 

2.5 

.7 

19030 

19105 

19130 

19156 

19181 

2.5 

.8 

19208 

19231 

19256 

19232 

19307 

2.5 

.9 

19332 

19357 

19382 

19407 

19432 

2.5 

21°.  o 

19457 

19482 

19507 

19532 

19557 

2.5 

.1 

19582 

19808 

19(531 

19656 

19681 

2.5 

.2 

19708 

19730 

19755 

19780 

19804 

2.5 

.3 

19823 

19854 

19378 

19903 

19927 

2.5 

.4 

19952 

19376 

20001 

20025 

20050 

2.5 

.5 

20074 

20098 

20123 

20147 

20172 

2.5 

.6 

20196 

20220 

20244 

20269 

20233 

2.4 

.7 

20317 

20311 

20365 

20389 

20413 

2.4 

.8 

20438 

20462 

20486 

20510 

20534 

2.4 

.9 

20558 

20581 

20605 

20629 

20653 

2.4 

22°  .0 

20677 

20701 

20725 

20748 

20772 

2.4 

.1 

20793 

20820 

20843 

20867 

20891 

2.4 

.2 

20914 

20938 

20962 

20985 

21009 

2.4 

.3 

21032 

21056 

21079 

21103 

21126 

2.4 

.4 

21150 

21173 

21196 

21220 

21243 

2.3 

.5 

21266 

21230 

21313 

21336 

21359 

2.3 

.6 

21383 

21408 

21429 

21452 

21475 

2.3 

.7 

21498 

21522 

21545 

21568 

21591 

2.3 

.8 

21614 

21637 

21660 

21683 

21706 

2.3 

.9 

21728 

21751 

21774 

21797 

21820 

23 

23°.  0 

21843 

21866 

21888 

21911 

21934 

2.3 

.1 

21957 

21979 

22002 

22025 

22C47 

2.3 

.2 

22070 

220:2 

22115 

22138 

22160 

2.3 

.3 

22183 

22205 

22228 

22250     22272 

2.2 

.4 

22295 

22317 

22340 

22362 

22:384 

2.2 

.5 

22407 

22429 

22451 

23474 

22496 

2.2 

.6 

22518 

22540 

22562 

22585 

22607 

2.2 

.7 

22629 

22351 

22673 

22695 

22717 

2.2 

.8 

22739 

22761 

227a3 

22805 

22827 

2.2 

.9 

22849 

22871 

S3893 

22915 

22937 

2.2 

24°  0 

22959 

22981 

23003 

23024 

23046 

2.2 

.1 

23088 

23090 

23111 

23133 

23155 

2.2 

.2 

23176 

23198 

23220 

23241 

23263 

2.2 

.3 

23205 

23306 

23328 

23349 

23371 

2.2 

.4 

23392 

23414 

234.35 

23457 

23478 

2.2 

.5 

23500 

23521 

23542 

23564 

23585 

2.1 

.6 

23608 

23628 

23649 

23670 

23()92 

2.1 

.7 

23713 

23734 

23755 

23776 

23798 

2.1 

g 

23819 

23840 

23861 

23882 

23903 

2.1 

.'9 

23924 

23945 

23966 

23987 

24008 

2  1 

TABLE  XV.-FOR  OBTAINING  BAROMETRIC  HEIGHTS  IN  FEET. 


Barom- 
eter. 
Inches 

0.00 

0.02 

0.04 

0.06 

0.08 

Diff  .  per 
.002  in. 

25°.0 
--.I 
.2 

24029 
24134 
24238 

24050 
24155 
24259 

24071 
24176 

24280 

24092 
24197 
24301 

24113 
24217 
24321 

2.1 
2.1 
2.1 

.3 

24342 

24363 

243*4 

24404 

24425 

2.1 

.4 

24446 

24466 

24487 

24508 

24528 

2.1 

.5 

24649 

24569 

24590 

24610 

24(531 

2.1 

.6 

24651 

24672 

24692 

24713 

24733 

2.0 

.7 

24754 

24774 

24794 

24815 

24835 

2.0 

.8 

24855 

24876' 

24896 

24916 

24937 

2.0 

.9 

24957 

24977 

24997 

25018 

25038 

2.0 

26°.  o 

25058 

25078 

25098 

25118 

25138 

2.0 

.1 

25159 

25179 

25199 

25219 

25239 

2.0 

.2 

25259 

25279 

25299 

25319 

25*39 

2.0 

.3 

25359 

25379 

25399 

25419 

25438 

2.0 

.4 

25458 

25478 

25498 

25518 

25538 

2.0 

.5 

25557 

25577 

23597 

25617 

25687 

2.0 

.6 

25656 

25676 

25696 

25715 

25786 

2.0 

.7 

2575;5 

25774 

25794 

25818 

25888 

2.0 

.8 

25853 

25872 

25892 

25911 

25931 

2.0 

/J 

25950 

25970 

25989 

26009 

26028 

2.0 

27°.  0 

26048 

26067 

26086 

26106 

26125 

1.9 

.1 

26145 

26164 

26183 

26203 

26222 

1.9 

.2 

26241 

245260 

26280 

26299 

26318 

1.9 

.3 

26337 

26&57 

26376 

26395 

26414 

1.9 

.4 

26433 

26452 

26472 

2(5491 

26510 

1.9 

.5 

26529 

26548 

26567 

26586 

26605 

1.9 

.6 

26624 

26643 

26662 

26681 

26700 

1.9 

7 

26719 

26738 

26757 

26776 

26795 

1.9 

.'8 

26813 

26832 

26851 

2-.  -870 

26889 

1.9 

.9 

26908 

26926 

26945 

26964 

26983 

1.9 

28°.0 

27001 

27020 

27039 

27058 

27076 

1.9 

.1 

27095 

27114 

£7132 

27151 

27169 

>   1-9 

.2 

27188 

27207' 

27225 

27244 

27262 

1.9 

.3 

27281 

27299 

27318 

27336 

27855 

1.8 

.4 

27373 

27392 

27410 

27429 

27447 

1.8 

.5 

27466 

27484 

27502 

27521 

27539 

1.8 

.6 

27557 

27576 

27594 

27612 

27631 

1.8 

.7 

27649 

27(567 

27685 

27704 

27722 

1.8 

.8 

27740 

27758 

27777 

27795 

27'813 

1.8 

.9 

27881 

27849 

27867 

27885 

27904 

1.8 

29°.  0 

27922 

27940 

27958 

27976 

27994 

1.8 

.1 

28012 

28030 

28048 

28066 

28084 

1.8 

.2 

28102 

28120 

28188 

28156 

28174 

1.8 

.3 

28192 

28209 

28227 

28245 

28263 

1.8 

.4 

23281 

28299 

28317 

28834 

28853 

1.8 

.5 

28370 

28388 

28405 

28423 

28441 

1.8 

.6 

28459 

28476 

28494 

28512 

28529 

1.8 

.7 

88547 

23565 

28582 

28600 

88618 

1.8 

.8 

28635 

28653 

28(570 

28688 

28706 

1.8 

.9 

28723 

28741 

28758 

28776 

28793 

1.8 

30°.  0 

28811 

28828 

28846 

28863 

28881 

1.8 

.1 

28898 

28915 

28933 

28950 

28968 

1.8 

.2 

23985 

2900.3 

29020 

29037 

29054 

1.7 

.3 

29072 

29089 

29106 

29124 

29141 

1.7 

.4 

29158 

29175 

20192 

29210 

29227 

1.7 

.5 

29244 

29261 

29278 

29296 

29313 

1.7 

.6 

29330 

29347 

29364 

29381 

25)398 

1.7 

7 

29416 

29433 

29450 

29467 

29484      1.7 

[fi 

29501 

29518 

29535 

29552 

29569      1.7 

.9 

29586 

29C03 

£9620 

29637 

29654   ,    1.7 

COS 


TABLE  XVI.— COEFFICIENT  OF  CORRECTION   FOR  TEMPERATURE. 


.+* 

t  +  t'-W 
900 

t+t' 

\t+t'  -  64° 

t  +  t' 

t  +  f  -  64° 

«+ 

, 

t  +  f  -  64° 

900 

900 

900 

20° 

.0489 

65° 

. 

.0011 

110° 

_|_ 

.0511 

155° 

.1011 

21 

— 

.0478 

66 

.0022 

111 

.0522 

156 

.1022 

23 

.0467 

67 

.0033 

IIS 

1  . 

.0688 

157 

.1033 

23 

.0456 

68 

.0044 

113 

.0544 

158 

.1044 

24 

.0444 

69 

.0056 

114 

ir 

.0556 

159 

.1056 

25 

.0433 

70 

.0067 

115 

.0567 

160 

.1067 

26 

.0422 

71 

.0078 

lie 

i 

.0578 

161 

.1078 

27 

.0411 

72 

.0089 

117 

.0589 

.1089 

28 

.0400 

73 

.0100 

11* 

\ 

.0600         163 

.1100 

29 

.0:389 

74 

.0111 

ni 

> 

.0611 

164 

.1111 

30 

.0378 

75 

.0122 

120 

-f 

.0622 

165 

.1122 

31 

— 

.0367 

76 

-4- 

.0133 

121 

.06133 

166 

+  .1133 

32 

.0356 

77 

.0144 

12* 

i 

.0644 

167 

.1144 

33 

.0344 

78 

.0156 

123 

.0656 

!  168 

.1156 

34 

.0333 

79 

.0167 

124 

l 

.0667 

169 

.1167 

35 

.0322 

80 

.0178 

125 

.0678 

170 

.1178 

36 

.0311 

81 

.0189 

126 

.0689 

171 

.1189 

37 

.0300 

82 

.0200 

127 

.0700 

172 

.1200 

38 

.0289 

83 

.0211 

1& 

\ 

.0711 

173 

.1211 

39 

.0278 

84 

.0222 

129 

.0722 

174 

.1222 

40 

.0267 

85 

.0233 

13( 

» 

-f 

.0733 

175 

.1233 

41 

_ 

.0256 

86 

+ 

.0244 

131 

.0744 

176 

+  .1244 

42 

.0244 

87 

.0256 

13$ 

! 

.0756 

177 

.1256 

43 

.0233 

88 

.0267 

133 

.0767 

J  178 

.1267 

44 

.0222 

89 

.0278 

134 

.0778 

179 

.1278 

45 

.0211 

90 

.0289 

13£ 

1 

.0789 

180 

.1289 

46 

.0200 

91 

.0300 

136 

.0800 

181 

.1300 

47 

.0189 

92 

.0311 

137 

.0811 

182 

.1311 

48 

.0178 

93 

.0322 

138 

.0822 

183 

.1322 

49 

.0167 

94 

.0333 

m 

.0833 

184 

.1333 

50 

_ 

.0156 

95 

.0344 

140 

_|_ 

.0844 

185 

.1344 

51 

.0144 

96 

+ 

.0356 

141 

.0856 

186 

-j-     1356 

52 

.0133    . 

97 

.0367 

142 

.0867 

187 

!l367 

53 

.0122 

98 

<* 

.0378 

143 

.0878 

188 

.1378 

54 

.0111 

99 

.0389 

144 

.0889 

189 

.1389 

55 

.0100 

100 

.0400 

145 

.0900 

190 

.1400 

56 

.0089 

101 

.0411 

146 

.0911 

191 

.1411 

57 

« 

.0078 

102 

.0422 

147 

.0922 

192 

.1422 

58 

.0067 

103 

.0433 

148 

.0933 

193 

.1433 

59 

.0056 

j  104 

.0444 

148 

.0944 

194 

.1444 

60 

.0044 

i  105 

.0456 

15C 

' 

-f 

.0956 

195 

.1456 

61 

_ 

.0033 

106 

_j_ 

.0467 

151 

.0967 

196 

+   .1467 

62 

.0022 

107 

.0478 

152 

.0978 

i  197 

.1478 

63 

.0011 

108 

.0489 

153 

.0'J&9 

198 

.1489 

64 

.0000 

109 

.0500 

154 

.1000 

199 

.1500 

1 

TABLE 

XVIL-CORRECTION  FOR  EARTH'S  CURVATURE  AND 

REFRACTION. 

§119. 

L° 

H° 

L° 

H° 

V 

H° 

1? 

H° 

L°       H° 

Miles      H° 

300 

.002 

|  1300 

.0.35  I 

8300 

.108 

asoo 

.223 

4300     .879 

1          .571 

400 

003 

1400 

.040 

2400 

.118 

3400 

.237 

4400     .397 

2         2.285 

500 

.005 

!   1500!   .046 

2500 

128 

3500 

.251 

4500     .415 

3         5.142 

600 

007 

1600     .052 

2600 

.139 

3600 

.266 

4600     .434 

4         9.141 

700 

.010 

1700;   .059  : 

2700 

.149 

3700 

.281 

4700     .453 

5       14.282 

800 

.013 

1800 

.066  , 

2800 

.161 

3800 

.298 

4800     .472 

6       20.567 

900 

.017 

191)0 

.074 

2900 

.172 

3900 

.312 

4900  |    .492 

7       27.994 

1000 

020. 

>  2000 

082  ! 

3000 

.184 

4000 

.328 

50001    .512 

8       36.563 

1100 
1200 

.025 
.030 

:  2100 

2200 

.090  j 
.099 

3100 
3200 

.197 
.210 

4100  j    ,345 
4200  i   .362 

5100  i    .533 
5200     .554 

9     i  46.275 
10       57.130 

009 


TABLE   XVIII.-COEFFICIENT   FOR   REDUCING  INCLINED  STADIA 
MEASUREMENTS  TO  THE  HORIZONTAL.    §  224. 


a 

0' 

10' 

20' 

30'       40' 

50' 

0° 

1.000000 

.999992 

.999967 

.999924 

.999865 

.999789 

1 

.99C696 

.999586 

.999459 

.999315 

.999154 

.998977 

2 

.998782 

.998571 

.998343 

.998098 

.997836 

.997557 

3 

.997261 

.996949 

.996619 

.996273 

.995910 

.995531 

4 

.995134 

.994721 

.994291 

.993844 

.993381 

.992901 

5 

.992404 

.991891 

.991360 

.990814 

.990250 

.989670 

6 

.989074 

.988461 

.987831 

.987185 

.986522 

.985843 

7 

.985148 

.984436 

.983708 

.982963 

.982202 

.981424 

8 

.980631 

.979821 

.978995 

.978152 

.977294 

.976419 

9 

.975528 

.974621 

.973698 

.972759 

.971804 

.970833 

10 

.969846 

.968843 

.967824 

.966790 

.965739 

.964673 

11° 

.96:3591 

.962494 

.961380 

.960252 

.959107 

.957948 

12 

.956772 

.955581 

.954375 

.953153 

.951916 

.950664 

13 

.949396 

.948113 

.946815 

.945.502 

.944174 

.942831 

14 

.941473 

.940100 

.938711 

.937309 

.935891 

.934459 

15 

.933011 

.931550 

.930073 

.928582 

.927077 

.925557 

16 

.924022 

.922474 

.920911 

.919334 

.917742 

.916137 

17 

.914517 

.912883 

.911236 

.909574 

.907899 

.906209 

18 

.904507 

.902790 

.901060 

.899:316 

.897558 

.895787 

19 

.894003 

.892206 

.890395 

.888571 

.886733 

.884883 

20 

.883020 

.881143 

.879254 

.877352 

.875437 

.873510 

21° 

.871569 

.869617 

.867652 

.865674 

.863684 

.861681 

22 

.859667 

.857(540 

.855601 

.853550 

.851487 

.849412 

23 

.847326 

.845227 

.843117 

.840996 

.838862 

.836718 

24 

.834561 

.832394 

.830215 

.828025 

.825825 

.823613 

25 

.821390 

.819156 

.816911 

.814656 

.812390 

.810113 

26 

.807826 

.805529 

.803221 

.800903 

.798575 

.796236 

27 

.793888 

.791529 

.789161 

.786783 

.784396 

.781998 

28 

.779591 

.777175 

.774749 

.772314 

.769870 

.767416 

29 

.764934 

.762483 

.760002 

.757513 

.755015 

.752509 

30 

.749994 

.747471 

.744939 

.742399 

.739850 

.737294 

31° 

.734729 

.732157 

.729577 

.726989 

.724393 

721790 

32 

.719179 

.716561 

.713935 

.711302 

.708662 

.706015 

33 

.703361 

.700700 

.698033 

.695358 

692677 

.689990 

34 

.687296 

.684595 

.681889 

.679176 

.676457 

.673733 

35 

.671002 

.668266 

.665524 

.662776 

660023 

.657264 

36 

.654500 

.651731 

.648957 

.646177 

.643393 

.640604 

37 

.637810 

.635011 

.632208 

.629401 

.626588 

623772 

38 
39 

.620952 
.603946 

.618127 
.601099 

.615299 
.598248 

.612466 
.595395 

.609630 
.592537 

.606790 
.589677 

40 

.586814 

.5&3948 

.581079 

.578207 

.575332 

.572455 

41° 

.569576 

.566694 

.563810 

.660924 

.558036 

.555145 

42 
43 

44 

.552253 

.534867 
.517438 

.549a59 
.531964 
.514530 

.546464 
.529061 
.511622 

.543567 
.526156 
.508714 

.540668 
.523251 
.505805 

.537768 
.520345 

.502897 

45 

.499988 

.497079 

.494170 

.491261 

.488353 

.485445 

310 


TABLE    XIX.  -  LOGARITHM    OF    COEFFICIENT    FOR    REDUCING    IN- 
CLINED STADIA  MEASUREMENTS  TO  THE  HORIZONTAL.     fc!>24. 


a 

0' 

10' 

2(X 

30' 

40'       50' 

0° 

i 

2 
3 
4 
5 

0.000000 
9.999868 

.999471 
.998809 
.997882 
.996689 

9.999996 
.999820 
.999379 
.998673 
.997701 
.996464 

9.999985 
.999765 
.999280 
.998529 
.997514 
.990232 

9.999967 
.999702 
.999173 
.998379 
.997318 
.995992 

9.999941 
.999683 

.999059 
.998220 
.997116 
.995745 

9.999908 
.999555 
.998938 
.998055 
.996906 
.995491 

6° 

7 
8 
9 
10 

9.995229 
.993501 
.991506 
.989240 
.986703 

9.994959 
.993187 
.991147 
.988836 
.986253 

9.994683 
.992866 
.990780 
.988424 
.985797 

9.994399 
.992537 
.990406 
.988005 
.985332 

9.994107 
.992201 
.990025 
.987579 

.'84860 

9.993808 
.991857 
.989636 
.987144 
.984:380 

11° 

JO 

13 
14 
15 

9.983893 
.980808 
.977447 
.973808 
.969887 

9.983398 
.980268 
.976860 
.973174 
.969206 

9.982895 
.979719 
.976265 
.972532 
.968517 

9.982385 
.979163 
.975663 
.971883 
.967820 

9.981867 
.978599 
.975052 
.971225 
.967116 

9.981342 
.978027' 
.974434 
.970560 
.966403 

16° 
17 

18 
19 
20 

9.965683 
.961192 
.956412 
.951339 
.945970 

9.964954 
.960415 
.955587 
.950465 
.945047 

9.964218 
.959631 
.954753 
.949583 
.944114 

9.963473 

.958838 
.953912 
.948692 
.943174 

9.962721 
.958087 
.958068 

.947793 
.942225 

9.961960 
.957229 
.952205 
.946886 
.941268 

21° 
22 
23 

24 
25 

9.940302 
.934330 
.928050 
.921458 
.914549 

9.939328 
.933305 
.926974 
.920329 
.913366 

9.938345 
.932271 
.925888 
.919191 
.912175 

9.937354 
.931229 
.924794 
.918044 
.910974 

9:936355 
.930178 
.923691 
.916888 
.909764 

9.935347 
.929119 
.922579 
.915723 
.908546 

26° 

27 
28 
29 
CO 

9.907318 
.899759 
.891867 
.883635 
.875058 

9.906081 
.898467 
.890519 
.882230 
.873594 

9.904835 
.897166 
.889161 
.880815 
.872121 

9.903580 

.895855 
.887794 
.879390 
.870637 

9.902316 
.894535 
.886417 
.877956 
.869144 

9.901042 

.893.206 
.885031 
.876512 
.867641 

31° 
32 
33 
34 
35 

9.866127 
.a56837 
.847178 
.837144 
.826724 

9.864604 
.855253 
.845532 
.835434 
.824949 

9.863071 
.858659 
.843876 
.833714 
.823163 

9.861528 
.852054 
.842209 
.&31982 
.821367 

9.859974 
.850439 
.840531 
.830240 
.819559 

9.858411 
.848814 
.838843 
.828488 
.817740 

36° 
37 
38 
39 
40 

9.815910 
.804691 
.793058 
.780998 
.768500 

9.814068 
.802781 
.791078 
.778946 
.766374 

9.812216 

.8008(50 
.789086 
.776882 
.764235 

9.810352 
.798927 

.787'082 
.774805 
.762083 

9.808476 
.796982 
.785066 
.772716 
.759919 

9.806589 
.795026 
.7830:38 
.770614 
.757742 

41° 
42 
43 
44 
45 

9.755552 
.742138 
.728246 
.713858 
9.698959 

9.753349 
.739857 
,725883 
.711411 
9.696425 

9.751133 
.737561 
.723506 
.708950 
9.693876 

9.748904 
.735253 
.721115 
.706474 
9.691313 

9.746662 
.732931 
.718710 
!  708988 
9.688734 

9.744407 
.730595 
.716291 
.701479 
9.686140 

TABLE  XX.-LENGTHS  OF  CIRCULAR  ARCS;  RADIUS    =  1. 


Sec. 

1   ,-   .  *:. 

| 
Length. 

Min. 

Length. 

1 
Deg. 

Length. 

Deg.  | 
I 

Length. 

1 

.0000048 

1 

.0002909 

1 

.0174533 

61 

1.0646508 

2 

.0000097 

2 

.0005818 

2 

.0349066 

62 

1.0821041 

3 

.0000145 

3 

.0008727 

3 

.0523599 

63 

1  .0995574 

4 

.0000194 

4 

.0011636 

4 

.0698132 

64  | 

1.1170107 

5 

.0000242 

5 

.0014544 

5 

.0872665 

65 

1.1344640 

6 

.0000291 

6 

.0017453 

6 

.1047198 

66 

1.1519173 

7 

.0000339 

7 

.0020362 

7 

.1221730 

67 

1.1693706 

8 

.0000388 

8 

.0023271 

8 

.1396263 

68 

1.1868289 

9 

.0000436 

9 

.0026180 

9 

.1570796 

69 

1.2042772 

10 

.0000485 

10 

.0029089 

10 

.1745329 

70 

1.2217305 

11 

.0000533 

11 

.0031998 

11 

.1919862 

71 

1.2391838 

12 

.0000582 

12 

.0034907 

12 

.£094395 

72 

1.2566371 

13 

.0000630 

13 

.0037815 

13 

.2268928 

73 

1.2740804 

14 

.0000679 

14 

.0040724 

14 

.2443461 

74 

1.2915436 

15 

.0000727 

15 

.004S633 

15 

.2617894 

75 

1.3089969 

16 

.0000776 

16 

.0046542 

16 

.2792527 

76 

1.3264502 

17 

.0000824 

17 

.0049451 

17 

.2967060 

77 

1.3439035 

18 

.0000873 

18 

.0052360 

18 

.3141593 

78 

1.8613568 

19 

.0000921 

19 

.0055269 

19 

.3316126 

79 

1.3788101 

20 

.0000970 

20 

.0058178 

20 

.3490659 

£0 

1.3962634 

21 

.0001018 

21 

.0061087 

21 

.3665191 

81 

1.4137167 

22 

.0001067 

22 

.0063995 

22 

.SfcS9724 

82 

1.4311700 

23 

.0001115 

23 

.0066904 

23 

.4014257 

83 

1.  44  J-  6233 

24 

.0001164 

24 

.0069813 

24 

.4188790 

84 

1.46fc07C6 

25 

.0001212 

25 

.0072722 

25 

.4368323 

85 

1.4885299 

26 

.0001261 

26 

.0075631 

26 

.4537856 

86 

1.6008fc82 

07 

.0001309 

27 

.0078540 

27 

.4712389 

i  87 

1.5184864 

28 

.0001357 

28 

.0081449 

28 

.4886922 

88 

1.58Efcfc97 

29 

.0001406 

29 

.0084358 

29 

.5061455 

89 

1.  £583430 

30 

.0001454 

30 

.0087266 

30 

.5235988 

90 

1.5707963 

31 

.0001503 

31 

.0090175 

31 

.5410521 

91 

1.  5882486 

32 

.0001551 

32 

.0093084 

32 

.5685054 

92 

1.C057U29 

33 

.0001600 

33 

.0095993 

33 

.5768E87 

93 

1.C2315G2 

34 

.0001648 

34 

.0098902 

34 

.5934119 

94 

1.640COG5 

35 

.0001(597 

35 

.0101811 

35 

.6108652 

95 

1.  €580028 

36 

.0001745 

36 

.0104720 

36 

.6283185 

96 

1.6755161 

37 

.0001794 

37 

.0107C29 

37 

.6457718 

97 

1.  6828694 

38 

.0001842 

38 

.0110538 

38 

.6632251 

98 

1.7104227 

39 

.0001891 

39 

.0113446 

39 

.6806784 

99 

1.72787CO 

40 

.0001939 

40 

.0116355 

40 

.6981317 

100 

1.7453293 

41 

.0001988 

41 

.0119264 

41 

.7155850 

101 

1.7627825 

42 

.0002036 

42 

.0122173 

42 

.7380883 

102 

1.7802358 

43 

.0002085 

43 

.0125082 

43 

.7504916 

103 

1.797C891 

44 

.0062133 

44 

.0127991 

44 

.7679449 

104 

1.8151424 

45 

.0002182 

45 

•  .0130900 

45 

.7858982 

105 

1.8825857 

46 

.00022430 

46 

.0133809 

46 

.8028515 

106 

1.8*600480 

47 

.0002279 

47 

.0136717 

47 

.8208047 

107 

1.867  £023 

48 

.0002327 

48 

.0139626 

48 

.8377580 

108 

1.8849556 

49 

.0002376 

49 

.0142535 

49 

.8552113 

•109 

1.9024089 

50 

.0002424 

50 

.0145444 

50 

.8726646 

110 

1.9168622 

51 

.0002473 

51 

.0148353 

51 

.8901179 

111 

1.9373155 

52 

.0002521 

52 

.0151262 

52 

.9075712 

112 

1.9547688 

53 

.0002570 

53 

.0154171 

53 

.9250245 

113 

1.9722221 

54 

.0002618 

54 

.0157080 

54 

.9424778 

114 

1.8886753 

55 

.0002666 

55 

.0159989 

55 

.9599311 

115 

2.C071286 

56 

.0002715 

56 

.0162897 

56 

.9773844 

116 

2.024£819 

57 

.0002763 

57 

.0165806 

57 

.9948377 

117 

2.042C852 

58 

.0002812 

58 

.0168715 

58 

1.0122910 

118 

2  0594885 

69 

.0002800 

1  59 

.0171624 

59 

1.0297443 

119 

2.0769418 

60 

.0002909 

60 

.0174533 

60 

1.0471976 

12J 

1  2.0943951 

TABLE  XXI. -MINUTES  IN  DECIMALS  OF  A  DEGREE 


< 

or 

10" 

15" 

1 

|   20" 

30" 

40"   !   45"     50"    ' 

0 

.00000 

00278 

.00417 

.00556 

.00833 

.01111  I  .01250   .01389   0 

1 

.01667 

.01944 

.02083 

.02222 

.02500 

.02778 

.02917   .03055  |  1 

2 

.03333 

.03611 

.03750 

.03889 

.04167 

.04444 

.04583   .04722   2 

3 

.05000 

.05278 

.05417 

.05556 

.05833 

.06111 

.06250   .06:389 

3 

4 

.06S67 

.06944 

.07083 

.07222 

.07500 

.07778 

.07917   .08056 

4 

5 

.08333 

.08611 

.08750  i  .08889 

.09167 

.09444 

.09583   .09722 

5 

6 

.10000 

.  10278 

.10417 

.10556 

.10833 

.11111 

.11250  |  .11389 

6 

7 

.11667 

.11944 

.12083 

.  12222 

.12500 

.12778   .12917 

.13056 

7 

8 

13333 

.13611 

.13;  50 

.13889 

.14167 

.14444 

.14583 

.14722 

8 

9 

.  .15000 

.15278 

.15417 

.15556 

.15833   .16111 

.16250 

.16389 

9 

10 

.16667 

.16944 

.17083 

.17222 

.175JO   .17778   .17917 

.18056 

10 

11 

.18333 

.18611 

.18750 

.18889 

.19167 

19444 

.19583 

.19722 

11 

12 

.20000 

.20278 

.20417 

.20556 

.20833 

.21111 

.21250 

.21389 

12 

13 

.21667 

.21944 

.22083  j  .22222 

.22500   .22778 

.22917  j  .23056 

13 

14 

.23333 

.23611 

.23750   .23889 

.24167 

.24444 

.24583  |  .24722 

14 

15 

.25000 

.25278 

.25417   .25556 

.25833 

.26111 

.26250   .26389 

15 

16 

.26667 

.26944 

.27083 

.27222 

.27500  |  .27778 

.27917 

.28056 

16 

17 

.28333 

.28611 

.28750 

.28889 

.29167   .29444 

.29583 

.29722 

17 

18 

.30000 

.30278 

.30417 

.30556 

.30833 

.31111 

.31250 

.31389 

18 

19 

.31667 

.31944 

.32083 

.32222 

.32500 

.32778 

.32917 

.33056  i  19 

20 

.33333 

.33611 

.33750 

.33389 

.34167 

.34444 

.34583 

.34722  20 

21 

.3.5000 

.35278 

.35417 

.35556 

.35833 

.36111 

.36250 

.36389  21 

22 

.36667 

.36944 

.37083 

.37222 

.37500   .37778 

."37917   .38056 

22 

23 

.38333 

.38611 

.38750 

.38889 

.39167 

.39444 

.395a3   .39722 

23 

24 

.40000 

.40278 

.40417 

.40556 

.40833 

.41111 

.41250   .41389 

24 

25 

.41667 

.41944 

.42083 

.42222 

.42500 

.42778 

.42917 

.43056 

25 

26 

.43333 

.43611 

.43750 

.43889 

.44167 

.44444 

.44583 

.44722 

26 

27 

.45000 

.45278 

.45417 

.45556 

.45833 

.46111 

.46250 

.46389 

27 

28 

.46667 

.46944 

.47083 

.47222 

.47500 

.47778 

.47917 

.48056 

28 

29 

.48333 

.48611 

.48750 

.48889 

.49167 

.49444 

.49583 

.49722 

29 

30 

.50000 

.50278 

.50417 

.5^556 

.50833 

.51111 

.51250 

.51389 

30 

31 

.51667 

.51944 

.52083 

.52222 

.52501 

.52778 

.E2917 

.530.56 

31 

32 

.53333 

.53611 

.53750 

.53889 

.54167 

54444 

.54583 

.54722 

32 

33 

.55000 

.55278 

.55417 

.55556 

.55833 

iseni 

.56250 

.56389 

33 

34 

.56667 

.56944 

.57083 

.57222 

.57500 

.57778 

.57917 

.58056 

34 

35 

.58-333 

.58611 

.58750 

.58889 

.59167 

.59444 

.59583 

.59722 

35 

36 

.60000 

.60278 

.60417 

.60556 

.60833 

.61111 

.61250 

.61389 

36 

37 

.61667 

.61944 

.62083 

.62222 

.62500 

.62778 

.62917 

.63056 

37 

38 

.63333 

.63611 

.63750 

.63889 

.64167 

.64444 

.64583 

.64722 

38 

39 

.65000 

.65278 

.65417 

.65556 

.65833 

.66111 

.66250 

.66389 

39 

40 

.66667 

.66944 

.67083 

.67222 

.67500 

.67778 

.67917 

.68056 

40 

41 

.68333 

.68611 

.68750 

.68889 

.69167 

.69444 

.69583 

•  .69722 

41 

42 

.70000 

.70278 

.70417 

.70556 

.708a3 

.71111 

.71250 

.71389 

42 

43 

.71667 

.71944 

.72083 

.72222 

.72500 

.72778 

.72917 

.73056 

43 

44 

.73333 

.73611 

.73750 

.73889 

.74167 

.74444 

.74583 

.74722 

44 

45 

.75000 

.75278 

.75417 

.75556 

.75833 

.76111 

.76250 

.76389 

45 

46 

.76667 

.76944 

.77083 

.77222 

.77500 

.77778 

.77917 

.78056 

46 

47 

.78333 

.78611 

.78750 

.78889 

.79167 

.79444 

.79583 

.79722 

47 

48 

.80000 

.80278 

.80417 

.80556 

.80833 

.81111 

.81250 

.81389 

48 

49 

.81667 

.81944 

.82083 

.82222 

.82500 

.82778 

.82917 

.83056 

49 

50 

.83*33 

.83611 

.83750 

.83889 

.84167 

.84444 

.84583 

.847'22 

50 

51 

.85000 

.85278 

.85417 

.85556 

.85833 

86111 

.86250 

.86389 

51 

52 

.86667 

.86944 

.87083 

.87222 

.87500 

.87778 

.87917 

.88056 

52 

53 

.88333 

.88611 

.88750 

.88889 

.89167 

.89444 

.89583 

.89722 

53 

51 

.90000 

.90278 

.90417 

.90556 

.90833 

.91111 

.91250 

.91389 

54 

55 

.91667 

.91944 

.92083 

.92222 

.92500 

.92778 

.92917 

.93056 

55 

56 

.93333 

.93611 

.93750 

.93889 

.94167 

.94444 

.945&3 

.94722 

56 

57 

.95000 

.95278 

.95417 

.95556 

.95833 

.96111 

.96250 

.96389 

57 

58 

.96667 

.96944 

.97083 

.97222 

.97500 

.97778 

.97917 

.98056 

58 

59 

.98333 

.98611 

.99750 

.98889 

.99167 

.99444 

.99583 

.99722 

59 

' 

0-     10" 

15" 

20" 

30" 

40" 

45" 

50" 

31  o 


TABLE  XXII.-  INCHES  IN   DECIMALS  OF   A  FOOT. 


.  ' 

In. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9  i  10 

11 

In. 

i 

0 

Foot 

.0833 

.1667 

.  25001.  £333 

.4167 

.5000 

.5833 

.6667 

.7500!  .8333 

.9167 

0 

1-32 

.0026 

.0859 

.1693 

.2526 

.3:359 

.4193 

.5026 

.5859 

.6693 

.75261.8359 

.9193 

1-32 

1-16 

.0052 

.088.") 

.1719 

.2552 

.3385 

.4219 

.5052 

.58851.6719 

.75521.8385 

.9219 

1-16 

3-32 

.0078 

.0911 

.1745 

.2578 

.3411 

.4245 

.5078 

.5911 

.6745 

.75781.8411 

.9245 

3-32 

1-8 

.0104 

.09:38 

.1771 

.2604 

.3438 

.4271 

.5104 

.5938 

.0771 

.7604L8438 

.9271 

1-8 

5-32 

.0130 

.0964 

.1797 

.2630 

.34641  .42971.  5130 

.5964 

.6797 

.7630  .8464 

.9297 

5-32 

3-16 

.0156 

.0990 

.1823 

.2650 

.3490 

.4323 

.5156 

.5990 

.6823 

.7656  .8490 

.9323 

3-16 

7-32 

.0182 

.1016 

.1849 

.2682 

.3516 

.4349 

.5182 

.6016  .6849 

.7682:.  8516 

.9349 

7-32 

l-l 

.0208 

.1042 

.1875 

.2708 

.3542 

.4375 

.5208 

.6042  .6875 

.7708*.  8542 

.9375 

1-4 

9-32 

.0234 

.1068 

.1901 

.2734 

.3568 

.4401 

.5234 

.6068 

.6901 

.7734!.8568 

.9401 

9-32 

5-16 

.0260 

.1094 

.  1927 

.  2760:  .3594i.  4427|.5260 

.60941.6927 

.7760!.  8594 

.9427 

5-16 

11-32 

.0280 

.1120 

.1953 

.2780 

.3020 

.4453 

.5286 

.6120 

.6953 

.7786  .8620 

.9453 

11-32 

3-8 

.0313 

.1146 

.  1979 

.2813 

.3646 

.4479 

.5313 

.6146 

.6979 

7813  !  .8646 

.9479 

3-8 

13-32 

.0339 

.1172 

.2005 

.2839 

.36721  .4505[  .5339|  .6172|  .7005 

.7839  .867'2 

.1-505 

13-32 

7-16 

0365 

.1198 

.2031  .2865 

.3698 

.4531 

.5365 

.6198 

.7031 

.7865  .8698 

.9531 

7-16 

15-32 

.0391  .1224 

.2057  .2891 

.3724 

.45571.5391 

.6224  .7057 

.  7891  :.  8724 

.9557 

15-32 

1-2 

.0417 

.1250 

.2083 

.2917 

.3750 

.4583 

.5417 

.6250 

.7083  .7917  .8750 

.9583 

1-2 

17-32 

.0443 

.1276 

.2109 

.2943J  .3776  .4609 

.5443 

.6276J.  7109 

.7943  .8776 

.9609 

17-32 

9-16 

.0469 

.1302 

.2135 

.2969 

.3802 

.46:35 

.5469 

.6302 

.7135 

.7969s  .8802 

.9635 

9-16 

19-32 

.0495 

.1328 

.2161 

.2995 

.3828 

.4661 

.5495 

.7161 

.7995!.8828 

.9661 

19-32 

5-8 

.0521 

.1354 

.2188 

.  3021  1.3854  .4688 

.55211.6354 

.7188 

.8021,.  8854 

.968- 

5-8 

21-32 

.0547 

.1380 

.2214 

.3047 

.3880 

.4714 

.5547 

.7214 

.8047;  .8880 

.9714 

21-32 

11-16 

.0573 

.1406 

.2240 

.3073 

.3906 

.4740 

.5573 

.7240 

.8073  .8906 

.9740 

11-16 

23-32 

.0599 

.1432 

.2260 

.3099 

.3932 

.4766 

.5599 

.7266 

.8099,.  8932 

.9766 

23-32 

3-4 

.0625 

.1458 

.2292 

.3125 

.3958 

.4792 

.5625 

.6458 

.7292 

.8125L8958 

.0792 

3-4 

25-32 

.0651 

.1484 

.2318  .3151 

!3984 

.4818 

.5651 

.6484 

.7318 

.8151  .8984 

.9818 

25-32 

13-16 
27-32 

7-8 

.0677 
.0703 
.0729 

.1510 
.1536 
.1563 

.2344  .3177 
.2370  .3203 
.2396  .3229 

.4010 
.4036 
.4063 

.4844 
.4870 
.4896 

.5677 
.5703 
.5729 

.6510  .7344 
.6536  .7370 
.6563.7396 

.81771.  90101.  9844  13-1  6 
.8203  .9036  .987027-32 
.8229  .9063!.  9896)  7-8 

29-32 

.0755 

.1589 

.24221.3255  .4089 

.4922 

.5755 

.65891.7422 

.8255  .9089 

.992229-32 

15-16 

.0781 

.1615 

.24481  .3281 

.4115 

.4948 

.5781 

.6615 

.7448 

.8281  .9115 

.9948 

15-16 

31-32 

.0807 

.1641 

.2474  .3307 

.4141 

.4974 

.5807 

.6641 

.7474 

.8307:  .9141 

.9974 

31-32 

,0 

1 

2 

3 

4 

5 

6 

7 

8 

9  i  10 

11 

:  .•'.'. 

814* 


TABLE  XXIII. —SQUARES,   CUBES,   SQUARE  ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

1 
Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.4142136 

1.2599210 

.500000000 

3 

9 

27 

1.7320508 

1.4422496 

.333333333 

4 

16 

64 

2.0000000 

1.5874011 

.250000000 

5 

25 

125 

2.2360680 

1.7099759 

.200000000 

6 

36 

216 

2  4494897 

1.8171206 

.  166666667 

7 

49 

343 

2.6457513 

1.9129312 

.  142857  143 

8 

64 

512 

2.8284271 

2.0000000 

.125000000 

9 

81 

729 

3.0000000 

2.0800837 

.111111111 

10 

100 

1000 

3.1622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166248             2.2239801 

.090909091 

12 

'144 

1728 

3.4641016             2.2894286 

.083333333 

13 

169 

2197 

3.6055513             2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.  -1101422 

.071428571 

15 

225 

3375 

3.8729833 

2.4662121 

.066666667 

16 

256 

4096 

4.0000000 

2.5198421 

.062500000 

17 

289 

4913 

4.1231056 

2.5712816 

.058823529 

18 

324 

5832 

4.2426407 

2.6207414 

.055555556 

It) 

361 

6859 

4.3588989 

2.6684016 

.052631579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

!tl 

441 

9261 

4.5825757 

2.7589243 

.047619048 

22 

484 

10C48 

4.6904158 

2.  80*0393 

.045454545 

23 

529   . 

12167 

4.7958315 

2.8438610 

.043478261 

24 

576 

13824 

4.8989795 

2.8844991 

.041666667 

25 

625 

15625 

5.0000000 

2.1)240177 

.04COOOOOO 

20 

676 

17576 

5.0990195 

2.9624060 

.038461538 

27 

729 

19683 

5.1961524 

3.000COOO 

.037037037 

28 

784 

21952 

5.2915026 

3.0365889 

.035714286 

29 

'     841 

24389 

5.3851648 

3.0723168 

.034482759 

30 

900 

27000 

5.4772256 

3.1072325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413806 

.032258065 

32 

1024 

32768 

5.6568542 

3.1748021 

.031250000 

33 

1089 

35937 

5.7445626 

3.2075343 

.030303030 

34 

1156 

39304 

5.8309519 

3.2396118 

.029411765 

35 

1225 

43875 

5.9160798 

3.2710663 

.028671429 

3(3 

1296 

46656 

6.0000000 

3.3019S72 

.027777778 

37 

1369 

50653 

6.0827625 

3.3322218 

.027027027 

38 

1444 

54872 

6.1644140 

3.3619754 

.026315789 

39 

1521 

59319 

6.2449980 

3.3912114 

.025641026 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1681 

68921 

6.4031242 

3.4482172 

.024390244 

42 

1764 

74088 

6.4807407 

3.476C266 

.023809524 

43 

18-49 

79507 

6.5574385 

3.5033981 

.023255814 

44 

1936 

85184 

6.6332496 

3.5303483 

.0227'27273 

45 

2025 

91125 

6.708.2039 

3.5568933 

.022222222 

46 

2116 

97336 

6.7823300 

3.5830479 

.021739130 

47 

2209 

103823 

6.8556546 

3.6088261 

.02127(5600 

48 

2304 

110592 

6.9282032 

3.6342411 

.020833333 

49 

2401 

117649 

7.0000000 

3.6593057 

.020408163 

50 

2500 

125000 

7.0710678 

3.6840314 

.020000000 

51 

2601 

132651 

7.1414284 

3.7084298 

.019607843 

52 

2704 

140608 

7.2111026 

3.7325111 

.019230769 

53 

2809 

148877 

7.2801099 

3.7562858 

.0188fJ7!i2r> 

54 

2916 

157464 

7.3484692 

3.7797631 

.018518519 

55 

3025 

166375 

7.4161985 

3.8029525 

.018181818 

56 

3136 

175616 

7.4833148 

3.8258624 

.017857143 

57 

3249 

185193 

7.5498344 

3.8485011 

.017543860 

58 

3364 

195112 

7.6157731 

3.8708766 

.017241379 

59 

3481 

205379 

7.6811457 

3.8929965 

.016949153 

60 

3600 

216000 

7.7459667 

3.9148676 

.016666667 

61 

3721 

226981 

7.8102497 

3.93(54972 

.016393443 

62 

3844 

238828 

7.8740079 

3.9578915 

.016129032 

UL5 


CUBE  ROOTS,   AND  RECIPROCALS. 


r 

No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

63 

3969 

250047 

7.9372539 

3.9790571 

.015873016 

64 

4096 

262144 

8.0000000 

4.0000000 

.015625000 

6,5 

4225 

274625 

8.0622577 

4.0207256 

.015:384615 

60 

4356 

287496 

8.1240384 

4.0412401 

.015151515 

67 

4489 

300763 

8.1853528 

4.0615480 

.01492,5373 

68 

4024 

314432 

8.2462113 

4.0816551 

.014705882 

69 

4761 

328509 

8.3066239 

4.1015661 

.014492754 

70 

4900 

343000 

8.3666003 

4.1212853 

.014285714 

71 

5041 

357911 

8.4261498 

4.1408178 

.014084507 

72 

5184 

373248 

8.4852814 

4.1601676 

.013888889 

73 

5329 

389017 

8.5440037 

4.1793390 

.013698630 

74 

5476 

405224 

8.6023253 

4.1983364 

.013513514 

75 

5625 

421875 

8.6602540 

4.2171633 

.013*33333 

76 

5776 

438976 

8.7177979 

4.2358236 

.013157895 

77 

5929 

456533 

•8.7749644 

4.2543210 

.012b870l3 

78 

6084 

474552 

8.8317609 

4.2726586 

.012820513 

79 

6241 

493039 

8.8881944 

4.2908404 

.012658228 

80 

6400 

512000 

8.9442719 

4.3088695 

.012500000 

81 

6561 

531441 

9.0000000 

4.3267487 

.012345679 

82 

6724 

551368 

9.0553851 

4.3444815 

.012195122 

83 

6889 

571787 

9.1104336 

4.3620707 

.012048193 

84 

7056 

592704 

9.1651514 

4.3795191 

.011904762 

85 

7225 

614125 

9.219.5445 

4.3988296 

.011764706 

86 

7396 

636056 

9.2736185 

4.4140049 

.011627907 

87 

7569 

658508 

9.3273791 

4.4310476 

.011494253 

83 

7744 

681472 

9.3808315 

4.4479602 

.01136:3636 

•  89 

7921 

704969 

9.4:339811 

4.4647451 

.011235955 

90 

8100 

723000 

9.4868330 

4.4814047 

.011111111 

91 

8231 

753571 

9.5393920 

4.4979414 

.010989011 

92 

8464 

778683 

9.5916630 

4.5143574 

.010369565 

93 

8849 

804357 

9.6436508 

4.5306549 

.010r52688 

94 

8836 

830584 

9.6953597 

4.5468359 

.010638298 

M 

9025 

857375 

9.7467943 

4.5629026 

.010526316 

96 

9216 

884736 

9.7979590 

4.5788570 

.010416667 

87 

9403 

912673 

9.8488578 

4.5947009 

.010309278 

98 

9504 

941192 

9.8994949 

4.6104363 

.010204082 

90 

9801 

970299 

9.9498744 

4.6260350 

.010101010 

ioa 

10000 

1000000 

10.0000000 

4.6415888 

.010000000 

101 

10201 

1030301 

10.0498756 

4.6570095 

.009900990 

102 

10404 

1061208 

10.0995049 

4.6723287 

.009303922 

103 

10609 

1092727 

10.1488916 

4.6875482 

.0097087:38 

104 

10816 

1124864 

10.1980390 

4.7026694 

.009615385 

105 

11025 

H678S5 

10.2469508 

4.7176940 

.009523810 

106 

11236 

1191016 

1Q.  2956301 

4.7326235 

.009433962 

107 

11449 

1225043 

10.3440804 

4.7474594 

.009345794 

108 

11664 

1259712 

10.3923048 

4.7622032 

.00:'259259 

109 

11881 

1295029 

10.4403065 

4.7768562 

.009174312 

110 

12100 

1331000   ;  10.488C885 

4.7914199 

.009090909 

111 

12321 

1367631 

10.5356538 

4.8058955 

.009009009 

112 

12544 

1404928 

10.5830052 

4.8202845 

.  0089285  il 

113 

12769 

1442897 

10.G301458 

4.8345881  ' 

.008849558 

114 

12996 

1481544 

10.  (770783 

4.848S076 

.008771930  ' 

115 

13225 

1520875 

10.7238053 

4.8629442 

.008695652 

116 

13456 

1560396 

10.7703296 

4.8769990 

.008620690 

117 

13689 

1601613 

10.  8166538 

4.8909732 

.003547009 

118 

13924 

1643032 

10.8627805 

4.9048681 

.008474576 

119 

14161 

1685159 

10.9087121 

4.9186847 

.008403361 

120 

14400 

1728000 

10.9544512 

4.9324249 

.008333333 

121 

14641 

1771561 

11.00.0000 

4.9460874 

.0032(54463 

188 

14884 

1815848 

11.0453610 

4.95967'57 

.008196721 

123 

15129 

1860867 

11.0905365 

4.9731898 

.008130081 

124 

15376 

1906624 

ll.ia>5287 

4.9866310 

.008064516 

31G 


TABLE  XXIII.— SQUARES,  CUBES,  SQUARE  ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

125 

15625 

1953125 

11.1803399 

5.0000000 

.008000000 

126 

15876 

2000376 

11.22497'22 

5.0132979 

.007936508 

127 

16129 

2048383 

11.2694277 

5.0265257 

.007874016 

128 

16384 

2097152 

11.3137085 

5.0396842 

.007812500 

129 

16641 

2146689 

11.3578167 

5.0527743 

.007751938 

130 

16900 

2197000 

11.4017543 

5.0657970 

.007692308 

131 

17161 

2248091 

11.4455231 

5.0787531 

.007633588 

132 

17424 

2299968 

11.4891253 

5.0916434 

.007575758 

133 

17689 

2352637 

11.5325626 

5.1044687 

.007518797 

134 

17956 

2406104 

11.5758369 

5.1172299 

.007462687 

135 

18225 

2460375 

11.6189500 

5.1299278 

.007407407 

136 

18496 

2515456 

11.6619038 

5.1425632 

.007352941 

137 

18769 

2571353 

11.7046999 

5.1551367 

.007299270 

138 

19044 

2628072 

11.7473401 

5.1676493 

.007246377 

139 

19321 

2685619 

11.7898261 

5.1801015 

.007194245 

140 

19600 

2744000 

11.8321596 

5.1924941 

.007142857 

141 

19881 

2803221 

11.8743421 

5.2048279     .007092199 

142 

20164 

2863288 

11.9163753 

5.2171034 

.007042254 

143 

20449 

2924207 

11.9582607 

5.2293215 

.006993007 

144 

20736 

2985984 

12.0000000 

5.2414828 

.006944444 

145 

21025 

3018625 

12.0415946 

5.2535879 

.006896552 

146 

21316 

3112136 

12.0830460 

5.2656374 

.006849315 

147 

21609 

3176523 

12.1243557 

5.2776321 

.006802721 

148 

21904 

3241792 

12.1655251 

5.2895725 

.006756757 

149 

22201 

8307949 

12.2065556 

5.3014592 

.006711409 

150 

22500 

3375000 

12.2474487 

5.3132928 

.006666667 

lol 

22801 

3442951 

12.2882057 

5.3250740 

.006622517 

152 

23104 

3511808 

12.3288280 

5.33G8033 

.006578947 

153 

23409 

3581577 

12.3693169 

5.3484812 

.006535948 

154 

23716 

3652264 

12.4096736 

5.3601084 

.006493506 

155 

24025 

3723875 

12.449899(5 

5.3716854 

.006451613 

156 

24336 

3796416 

12.48999GO 

5.3832126 

.006410256 

157 

24649 

3869893 

12.5299641 

5  3946907 

.006369427 

158 

24964 

3944312 

12.5698051 

5.4061202 

.006329114 

159 

25281 

4019679 

12.6095202 

5.4175015 

.006289308 

160 

25600 

4096000 

12.6491106 

5.4288352 

.006250000 

101 

25921 

4173281 

12.6885775 

5.4401218 

.006211180 

162 

26244 

4251528 

12.7279221 

5.4513618 

.006172840 

163 

26569 

4330747 

12.7671453 

5.4625556 

.006134969 

164 

26896 

4410944 

12.8062485 

5.4737037 

.006097561 

165 

27225 

4492125 

12.8452326 

5.4848066 

.006060606 

166 

27556 

4574296 

12.8840987 

6.4958647 

.006024096 

167 

27889 

4657463 

12.9228480 

5.5068784 

.005988024 

168 

28224 

4741632 

12.9614814 

5.5178484 

.005952381 

169 

28561 

4826S09 

13.0000000 

5.5287748 

.005917160 

170 

28900 

4913000 

13.03^4048 

5.5396583 

.0058821353 

171 

29241 

5000211 

13.0766968 

5.5504991 

.005847953 

172 

29584 

5088448 

13.1148770 

5.5612978 

.005813953 

173 

29929 

5177717 

13.  1529464 

5.5720546 

.005780347 

174 

30276 

5268024 

13.1909060 

5.5827702 

.005747126 

175 

30625 

5359375 

13.2287566 

5.5934447 

.005714286 

176 

30976 

5451776 

13.2664992 

5.6040787 

.005681818 

177 

31329 

5545233 

13.3041347 

5.6146724 

.005649718 

178 

31684 

5639752 

13.3416641 

6.6252263 

.005617978 

179 

32041 

5735339 

13.3790882 

5.6357408 

.005586592 

180 

32400 

5832000 

13.4164079 

5.6462162 

.005555556 

181 

32761 

5929741 

13.4536240 

5.6566528 

.005524862 

182 

33124 

6028568 

13.4907376 

5.6670511 

.005494505 

183 

33489 

6128487 

13.5277493 

5  6774114 

.005464481 

1&4 

33856 

6229504 

13.5646600 

5.6877340 

.005434783 

185 

84225 

6331625 

13.6014705 

5.6980192 

.005405405 

186 

34596 

6434856 

13.6381817 

6.7082675 

.005376344 

317 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots.   Reciprocals. 

187 

34969 

6539203 

13.6747943 

5.7184791 

.005347594 

188 

35344 

6644672 

13.7113092 

6.7286543 

.005319149 

189 

35721 

6751269 

13.7477271 

5.7387936 

.005291005 

190 

36100 

6859000 

13.7840488 

6.7488971 

.005263158 

191 

36481 

6967871 

13.8202750 

6.7588652 

.005235602 

192 

36864 

7077888 

13.8564065 

5.7669982 

.005208333 

193 

37249 

7189057 

13.8924440 

5.7789966 

.005181347 

194 

37636 

7301384 

13.9283883 

5.7889604 

.005154639 

195 

38025 

7414875 

13.9642400 

5.7988900 

.005128205 

196 

38416 

7529536 

14.0000000 

5.8087857 

.005102041 

197 

38809 

7645373 

14.0356688 

5.8186479 

.005076142 

198 

39204 

7762392 

14.0712473 

5.8284767 

.005050505 

199 

39601 

7880599 

14.1067360 

5.8382725 

.005025126 

200 

40000 

8000000 

14.1421356 

6.8480355 

.005000000 

201 

40401 

8120601 

14.1774469 

5.8577660 

.004975124 

202 

40804 

8242408 

14.2126704 

6.8674643 

.004950495 

203 

41209 

8365427 

14.2478068 

5.8771307 

.004926108 

204 

41616 

8489664 

14.2828569 

5.8867653 

.004901961 

205 

42025 

8615125 

14.3178211 

5.8963685 

.004878049 

206 

42436 

8741816 

14.3527001 

5.9059406 

.004854369 

207 

42849 

8869743 

14.3874946 

5.9154817 

.0048£0918 

208 

43264 

8998912 

14.4222051 

5.9249921 

.004807692 

209 

43681 

9129329 

14.4568323 

5.9344721 

.004784689 

210 

44100 

9261000 

14.4913767 

5.9439220 

.004761905 

211 

44521 

9393931 

14.5258890 

5.8533418 

.004739336 

212 

44944 

9528128 

14.5C02198 

5.9627320 

.004716981 

213 

45369 

9663597 

14.5945195 

5.9720926 

.004694836 

214 

45796 

9800344 

14.6287388  !   5.8814240 

.004672897 

215 

46225 

9938375 

14.6628783 

5.9907264 

.004651163 

216 

46656 

101)77696 

14.6969385 

6.0COOCOO 

.004629630 

217 

47089 

10218313 

14.7309199 

6.CCS2450 

.004608295 

218 

47524 

10360232 

14.7648231 

6.0184617 

.004587156 

219 

47961 

10503459 

14.7986486 

6.0276502 

.C04E66210 

220 

48400 

10048000 

14.8S23970 

6.0368107 

.004545455 

221 

48841 

10793861 

14.8660687 

6.0459435 

.004524887 

222 

49284 

10941048 

14.8996644 

6.0250489 

.  004504  Ea5 

223 

49729 

11089567 

14.9331845 

6.C641270 

.004484305 

224 

50176 

11239424 

14.9666295 

6.0731779 

.004464286 

225 

50625 

11390625 

15.0000000 

6.C8S2020 

.004444444 

226 

51076 

11543176 

15.0332964 

6.0911994 

.004424779 

227 

51529 

11697083 

15.0665192 

6.1C01702 

.004405286 

228 

51984 

Iia52352 

15.0996689 

6.1091147 

.004385965 

229 

52441 

12008989 

15.1327460 

6.1160332 

.004366812 

230 

52900 

12167000 

15.1657509 

6.1269257 

.004347826 

231 

53361 

12326391 

15.1986842 

6.1357924 

.004329004 

.  232 

53824 

12487168 

15.2315462 

6.1446337 

.004310345 

233 

54289 

12649337 

15.2643375 

6.1  £34495 

.004291845 

234 

54756 

12812904 

15.2970585 

6.1622401 

.004273504 

235 

55225 

12977875 

15.3297097 

6.1710058 

.004255319 

236 

55696 

13144256 

15.3622915 

6.1797466 

.004237288 

237 

56169 

13312053 

15.3948043 

6.1884628. 

.004219409 

238 

56644 

13481272 

15.4272486 

6.1971544 

.004201681 

239 

57121 

13651919 

15.4596248 

6.2058218 

.004184100 

240 

57600 

13824000 

15.4919334 

6.2144650 

.004166667 

241 

58081 

13997521 

15.5241747 

6.2230843 

.C04149378 

242 

58564 

14172488 

15.5563492 

6.2316797 

.0041£2231 

243 

59049 

14348907 

15.5884573 

6.2402515  ' 

.004115226 

244 

59536 

14526784 

15.6204994 

6.2487998 

.C040C8261 

245 

60025 

14706125 

15.6524758 

6.2573248 

.004081  6£3 

246 

60516 

14886936 

15.6843871 

6.2658266 

.004C65041 

247 

61009 

15069223 

15.7162336 

6.2743054 

.004048583 

248 

61504 

15252992 

15.7480157 

6.2827613 

.004032258 

318 


TABLE  XXIII.— SQUARES,  CUBES,  SQUARE  ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Roots. 

Reciprocals. 

249 

62001 

15438249 

15.7797338 

6.2911946 

.004016064 

250 

62500 

15625000 

15.8113883 

6.2996053 

.004000000 

13.31 

63001 

15813251 

15.8429795 

6.30799:35 

.0039840(54 

2&1 

63504 

16003008 

15.8745079 

6.3163596 

.003968254 

988 

6400J 

16194277 

15.9059737 

6.3247035 

.003952569 

254 

64516 

16387064 

15.9373775 

6.3330256 

.003937008 

355 

65025 

16581375 

15.9687194 

6.3413257 

.003921569 

255 

65536 

16777216 

16.0000000 

6.3496042 

.003906250 

257 

66049 

16974593 

16.0.J12195 

6.3578611 

.003891051 

.258 

66564 

17173512 

16.0623784 

6.3600968 

.00:3875969 

259 

67081 

17373979 

16.0934769 

6.3743111 

.00:3861004 

260 

67600 

17576000 

16.1245155 

6.3825043 

.003846154 

261 

68121 

17779581 

16.1554944 

6.3906765 

.00:3831418 

263 

68644 

17984723 

16.1864141 

6.3988279 

.003816794 

263 

69169 

18191447 

16.2172747 

6.4069585 

.003802281 

264 

69696 

18399744 

16.2480768 

6.4150687 

.003787879 

265 

70225 

18(509625 

16.2788206 

6.4231583 

.003773585 

266 

70756 

18821096 

16.3095064 

6.4312276 

.003759398 

267- 

71289 

19034163 

16.3401346 

6.4392767 

.0037  45318 

268 

71821 

19248832 

16.37W055 

6.447:3057 

.003731343 

269 

72361 

19465109 

16.4012195 

6.4553148 

.003717472 

270 

72900 

19683000 

16.4316767 

6.4633041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003690037 

27'2 

73984 

20123648 

16.4924225 

6.4792236 

.003676471 

273 

7452  ) 

20316417 

16.5227116 

6.4871541 

.003663004 

27'4 

750  io 

2057'0824 

16.5529454 

6.4950653 

.003649635 

275 

75625 

20796875 

16.5831240 

6.5029572 

.003636364 

276 

76176 

21024576 

16.6132477 

6.5108300 

.003623188 

277 

76729 

21258933 

16.6433170 

6,5186839 

.003610108 

278 

.  77284 

21484952 

16.6733320 

6.5265189 

.003597122 

279 

77841 

21717639 

16.7032931 

6.5343351 

.003584229 

280 

78400 

21952000 

16.7332005 

6.5421326 

.003571429 

281 

78961 

22188041 

16.7630546 

6.5499116 

.003558719 

282 

79524 

22425768 

16.7928556 

6.5576722 

.003,546099 

283 

80089 

22665187 

16.8226038 

6.5654144 

.00:35:33569 

284 

80656 

22906304 

16.8522995 

6.5731385 

.003521127 

285 

81225 

23149J25 

16.8819430 

6.5808443 

.003508772 

286 

81796 

2*393656 

16.9115345 

6.5885323 

.003496503 

287 

82369 

23639903 

16.9410743 

6.5962023 

.003484321 

288 

82944 

23887872 

16.9705627 

6.60:38545 

.003472222 

289 

83521 

24137569 

17.0000000 

6.6114890 

.003460208 

200 

84100 

24389000 

17.0293864 

6.6191060 

.003448270 

291 

84681 

24642171 

7.0587221 

6.6267054 

.0034:36426 

292 

85264 

24807088 

7.0880075 

6.6342874 

.00:3424658 

21)3 

85849 

25153757 

7.1172428 

6.6418522 

.00:3412969 

294 

86433 

23412184 

7.1464282 

6.649399S 

.003401361  . 

295 

87025 

25672375 

7.1755640 

6.6569302 

.00:3389831 

296 

87616 

23934336 

7.2046505 

6.6644437 

.003378378 

297 

88209 

26198073 

7.2336879 

6.6719403 

.  003367033 

298 

88804 

26463592 

7.2626765 

6.6794200 

.003:355705 

299 

89401 

2673089;) 

7.2916165 

C.  6868831 

.003344182 

300 

90000 

27000000 

7.3205081 

G.  4943295 

.003333333 

301 

90601 

27270901 

7.:>493516 

6.7017593 

.00:3:322259 

302 

91204 

27543608 

7.37'81472 

6.7091729 

.003:311258 

303 

91809 

27818127 

7.4068952 

6.7165700 

•  .0033103:30 

304 

92416 

28094464 

7.4355958 

6.7239508 

.003289474 

305 

93025 

:*  28372625 

7.4642492 

6.7313155 

.003278689 

306 

93636 

'••28052616 

7.4928557 

6.7386641 

.003267974 

307 

94249 

28934443 

7.5214155 

6.7459967 

.0032.57329 

308 

94864 

29218112 

7.5499288 

6.7533134 

.00:3246753 

.  309 

95481 

29503629 

7.5783958 

6.7606143 

'  .003236246 

310 

96100 

29791000 

17.6068169 

6.7678995 

'  .003225806 

819 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Koots. 

Cube  Root  8. 

Reciprocals. 

311 

96721 

30080231 

17.6351921 

6.7751690 

.003215434 

312 

97344 

30371328 

17.0035217 

6.7824229 

.003205128 

313 

97909 

30604297 

7.6918000 

6.7'890013 

.0031948b8 

314 

98596 

30959144 

7.7200451 

6.71/08844 

.003184713 

315 

99223 

31255875 

7.7482393 

6.8U40921 

.003174603 

310 

99850 

31554496 

7.7763888 

6.8112847 

.003164557 

317 

100489 

31855013 

7.80449:38 

6.8184620 

.003154574 

318 

101124 

'  32157432 

7.8325545 

6.8256242 

.003144054 

319 

101701 

32461759 

17.8005711 

6.8327714 

.003134790 

320 

102400 

32768000 

17.8885438 

6.8399037 

.003125000 

321 

103041 

33076161 

17.9104729 

6.8470213 

.003115265 

p 

103084 

33386248 

17.9443584 

6.8541240 

.003105590 

828 

104329 

33698267 

17.9722008 

6.8612120 

.003095975 

324 

104976 

34012224 

18.0000000 

6.8682855 

.003086420 

325 

105625 

34328125 

18.0277564 

6.8753443 

.003076923 

338 

100270 

34645976 

18.0554701 

6.8823888 

.003067485 

327 

100929 

349(55783 

18.0831413 

6.8894188 

.003058104 

328 

107584 

35287552 

18.1107703 

6.8964345 

.003048780 

329 

108241 

35611289 

18.1383571 

6.9034359 

.003039514 

330 

108900 

35937000 

18.1659021 

6.9104232 

.003030303 

331 

109501 

3620401)1 

18.1934054 

6.917'3964 

.003021148 

332 

110224 

30594368 

18.2208072 

6.9243556 

.003012048  ' 

333 

110389 

30J20037 

18.2482870 

6.931:3008 

.003003003 

334 

111556 

37259704 

18.2756669 

6.9382321 

.002994012 

335 

118225 

37595375 

18.3030052 

6.9451490 

.002985075 

330 

118896 

37933056 

18.3303028 

6.9520533 

.00297'6190 

337 

113569 

3827'2753 

18.3575598 

6.9589434 

.002907359 

338 

114244 

38614472 

18.3847763 

6.9658198 

.002958580 

339 

114921 

38958219 

18.4119526 

6.9726826 

.002949853 

340 

115000 

39304000 

18.4390889 

6.9795321 

.002941176 

*  341 

1  16281 

39651821 

18-.4661853 

6.9803081 

.002932551 

342 

110904 

40001688 

18.4932420 

6.9931806 

.002923977 

343 

117049 

4035:3007 

18.5202592 

7.0000000 

.002915452 

344 

118330 

40707584 

18.5472370 

.0007962 

.002906977 

315 

119025 

41001302;") 

18.5741756 

0135791 

.002b98551 

340 

119710 

41421736 

18.6010752 

.0203490 

.002890173 

847 

12040'.) 

41781923 

18.027'9360 

.0271058 

.002881844 

348 

121104 

42144192 

18.6547581 

.0338497 

.002873503 

349 

121801 

42508549 

18.0815417 

.040:;806 

.002865330 

"50 

122500 

42875000 

18.  7'0828G9 

.0472987 

.002a57143 

351 

123201 

43243551 

18.7349940 

.0540041 

.00284<>00:3 

352 

123904 

43614208 

18.76166:30 

.0606967 

.002840909 

853 

124009 

43980977 

18.7882942 

.0673767 

.C02832h61  ' 

354 

125310 

44361864 

18.8148877 

.0740440 

.002824859 

355 

120025 

44738875 

18.8414437 

.0806988 

.002810901 

350 

120730 

45118016 

18.8679023 

.0873411 

.002808989 

357 

127449 

45499293 

18.8944436 

.0939709 

.002801120 

358 

128164 

45882712 

18.9208879 

.1005885 

.002793296 

359 

128881 

46268279 

18.9472953 

.1071937 

.002785515 

300 

129000 

46656000 

18.9736660 

.1137'86f> 

.002777778 

3ffl 

130321 

47045881 

19.0000000 

.1203674 

.002770083 

:Jtu 

131044 

47437928 

19.0262976 

.1269360 

.002762431 

383 

131709 

4?832147 

19.0525589 

.1334925 

.002754821 

304 

132490 

48228544 

19.0787840 

.1400370 

.002747253 

305 

133225 

48627125 

19.1049732 

.1465695 

.0027397'26 

300 

133950 

49027896 

19.1311265 

.15:30901 

.  002732240 

307 

134689 

49430863 

19.1572441 

.1595988 

.00272479.6 

308 

135424 

49830032 

19.1833261 

7.1660957 

.002717391 

309 

130101 

50243409 

19.2093727 

7.1725809 

.002710027 

370 

136900 

50653000 

19.2353841 

7.1790544 

.002702703 

371 

137641 

51004811 

19.2813603 

7.1855162 

.002695418 

372 

138384 

51478848 

19.2873015 

7.1919663 

.00268S172 

320 


TABLE  XXIII.-SQUARES,  CUBES,  SQUARE  ROOTS, 


No. 

Squares. 

nnhM      Square 
Cubes.      Koot& 

Cube  Roots.   Reciprocals. 

373 

139129 

51895117 

19.3132079     7.1984050      .002680965 

374 

139876 

62313624 

19.3390796 

7.2048322      .002673797 

375 

140625     52734375 

19.3649167 

7.2112479      .00266(5667 

376 

141376 

53157376 

19.3907194 

7.2176522      .002659574 

377 

142129 

53582633 

19.4164878 

7.2240450 

.002652520 

&78 

'  142884 

54010152 

19.4422221  !   7.2304268 

.  002645503 

379 

143641 

54439939 

19.4679^23     7.2367972 

.002638522 

380 

144400 

54872000 

19.4935887     7.2431565 

.002631579 

381 

145161 

55306341 

19.5192213  i   7.2495045 

.002624672 

38-3 

145924 

55742968 

19.5448203 

7.2558415 

.002617801 

383 

146689 

56181887 

19.5703858 

7.2621675 

.002610966 

384 

147456 

56623104 

19.5959179 

7.2684824 

.002604167 

385 

148225 

57066625 

19.6214169 

7.  2747864 

.002597403 

386 

148996 

57512456 

19.6468827 

7.2810794 

.002590674 

387 

149769 

57960603 

19.6723156 

7.2873617 

.002583979 

388 

150544 

58411072 

19.6977156 

7.2936330 

.002577320 

389 

151321 

58863869 

19.7230829 

7.2998936 

.002570894 

390 

152100 

59319000 

19.7484177 

7.3061436 

.002564103 

391 

152881 

59776471 

19.7737199  i   7.3123828 

.002557545 

392 

153664 

60236288 

19.7989899  j   7.3186114 

.002551020 

393 

154449 

60698457 

19.8242276 

7.3248295 

.002544529 

394 

15523G 

61162984 

19.8494332 

7.3310369 

.002538071 

895 

156025 

61629875 

19.8746069 

7.3372339 

.002531646 

396 

156816 

62099136 

19.8997487 

7.34S4205 

.OOS525253 

397 

157609 

62570773 

19.9248588 

7.3495966 

.002518892 

398 

158404 

63044792 

19.9499373 

7.3557624 

.002512563 

399 

159201 

63521199 

19.9749844 

7.3619178 

.002506266 

400 

160000 

64000000 

20.0000000 

7.3680630 

.002500000 

401 

160801 

64481201 

20.0249844 

7.3741979 

.002493706 

402 

161604 

64964808 

20.0499377 

7.3803227 

.002487562- 

403 

162409 

65450827     20.0748599 

7.3864373 

.002481390 

404 

163216 

65939264     20.0997512 

7.3925418 

.002475248 

405 

164025 

66430125 

20.1246118     7.3986863 

.002469136 

406 

164836 

66923416 

20.1494417     7.4047206 

.002463054 

407 

165649 

67419143 

20.1742410     7.4107950 

.002457002 

408 

166464 

67917312 

20.1990099     7.4168595 

.002450980 

409 

167'281 

68417929 

£0.2237484     7.4229142 

.002444988 

410 

168100 

68921000 

20.2484567     7.4289589 

.002439024 

411 

168921 

69426531 

20.2731349     7.4349938 

.  0024:33090 

412 

169744 

69934528 

20.2977831  j   7.4410189 

.002427184 

413 

170569 

70444997 

20.3224014  !   7.4470342 

.002421308 

414 

171396 

70957944 

20.3469899     7.4530399 

.002415459 

415 

172225 

71473375 

20.3715488    7.4590359 

.002409639 

416 

173056 

71991296 

20.3960781     7.4650223 

.002403846 

417 

173889 

72511713 

20.4205779     7.4709991 

.002398082 

418 

174724 

73034632 

20.4450483     7.4769664 

.002392344 

419 

175561 

73560059 

20.4694895     7.4829242 

.002386635 

430 

176400 

74088000 

20.4939015 

7.4888724 

.002380952 

421 

177241 

74618461 

20.5182845 

7.4948113 

.002375297 

422 

178084 

75151448 

20.5426386 

7.5007406 

.008369668 

423 

178929 

75686967 

20.5669638 

7.5066607 

.002364066 

424 

179776 

76225024 

20  5912603 

7,5125715 

.002358491 

425 

180625 

76765685 

20.6155281 

7.5184730 

.002352941 

426 

181476 

77308776 

20.6397674 

7.5243652  " 

.002347418 

427 

182329 

T7&54483 

20.6639783 

7.5302482 

.002341920 

428 

183184 

78402752 

20.6881609 

7.5361221 

.002336449 

429 

184041 

78953589 

20.7123152 

7.5419867 

.002331002 

430 

184900 

79507000 

20.7364414 

7.5478423 

.002325581 

431 

185761 

8C062991 

20.7605395 

7.5536888 

.002320186 

432 

186624 

80621568 

20.7846097 

7.5595263 

.002314815 

433 

187489 

81182737 

20.8086520  !   7.5653548 

.002309469 

434 

188356 

81746504 

20.8826667  i   7.5711743 

.002304147 

321 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

'Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

435 

189225 

82312875 

20.8566536 

7.5769849 

.002298851 

436 

190036 

82881856 

20.8806130 

7.5827865 

.002293578 

437 

190969 

83453453 

20.9045450 

7.5885793 

.002258330 

438 

191844 

8402,672 

20.9284495 

7.5943633 

.002283105 

439 

192721 

84604519 

20.9523268 

7.6001385 

.OJ2277904 

440 

193600 

85184000 

20.9761770 

7.6059049 

.032272727 

441 

194481 

85766121 

21.0000000 

7.6116626 

.002267574 

442 

195364 

86350888 

21.0237960 

7.6174116 

.002262443 

443 

196249 

86938307 

21.0475652 

7.6231519 

.002257336 

444 

197136 

87528:384 

21.0713075 

7.0288837 

.002252252 

445 

198025 

88121125 

21.0350231 

7.63460G7 

.002247191 

446 

198916 

88716536 

21.1187121 

7.6403213 

.002242152 

447 

199809 

89314623 

21.1423745 

7.6460272 

.002237136 

448 

200704 

89915392 

21.1660105 

7.6517247 

.002232143 

449 

201601 

90518849 

21.1896201 

7.  657413  J 

.002227171 

450 

202500 

91125000 

21.2132034 

7.6630943 

.002222222 

451 

203401 

91733851 

21.2367608 

7.6687665 

.002217295 

452 

204301 

92345438 

21.2602916 

7.6744303 

.002212389 

453 

205203 

92959677 

21.2837967 

7.6800857 

.002207506 

454 

206116 

93576664 

21.3072758 

7.6857323 

.002202643 

455 

207025 

94196375 

21.3307230 

7.6913717 

.002197802 

456 

207936 

94818816 

21.3541565 

7.6970023 

.002192932 

457 

203849 

95443993 

21.3775583 

7.7026246 

.002188184 

458 

203764 

96071912 

21.4009346 

7.7082388 

.002183406 

459 

210881 

96702579 

21.4242853 

7.7138443 

.002178649 

460 

211600 

97336000 

21.4476103 

7.7194423 

.002173913 

461 

212521 

97972181 

21.4703106 

7.7250325 

.002169197 

462 

213444 

98611128 

21.4941853 

.7336141 

.002164502 

4(53 

214369 

99252847 

21.5174348 

.7361877 

.002159827 

464 

215236 

99897344 

21.5406592 

.7417532 

.002155172 

485 

216225 

100544625 

21.5638587 

.7473109 

.002150538 

466 

217156 

101  194636 

21.5870331 

.7528606 

.002145923 

4(57 

218039 

101847563 

21.6101828 

.7584023 

.002141328 

468 

219024 

102503232 

21.6333077 

.7639361 

.002136752 

469 

219931 

103161709 

21.6564078 

.7694620 

.002132196 

470 

220900 

103823000 

21.6794834 

.7749801 

.002127660 

471 

221841 

104487111 

21.7025344 

.7804904 

.002123142 

472 

222784 

105154048 

21.7'255610 

.7859923 

.002118644 

473 

223729 

105823817 

21.7485632 

.7914875 

.002114165 

474 

221676 

103496424 

21.7715411 

.7-369745 

.002109705 

475 

225625 

107171875 

21.7944947 

.8024538 

.002105263 

476 

226576 

107850176 

21.8174242 

.8079254 

.002100840 

477 

227529 

103531333 

21.8403297 

.8133892 

.002096436 

478 

22S484 

109215352 

21  8632111 

.8188456 

.002092050 

479 

229441 

109902239 

21.8860686 

.8242942 

.002087683 

480 

230400 

110592000 

21.9089023 

.8297353 

.002088333 

481 

231361 

111284641 

21.9317122 

.8:351688 

.002079002 

482 

232324 

111980168 

21.9544984 

.8405949 

.002074689 

483 

233289 

112678587 

21.9772610 

.8460134 

.002070393 

484 

234256 

11:3379904 

22.0000000 

.8514244  . 

.002066116 

485 

235225 

114084125 

22.0227155 

.8568281 

.002061850 

486 

236196 

114791256 

22.0454077 

.8622242 

.002057613 

487 

237169 

115501303 

22.0680765 

.8676130 

.002053388 

488 

238144 

116214272 

22.09072.20 

.8729844 

.002049180 

489 

239121 

116930169 

22.11&3444 

.8783684 

.002044990 

490 

240100 

117649000 

22.1359436 

.8837352 

.002040316 

491 

241081 

118370771 

22.1585198 

.8890946 

.002036660 

492 

242064 

119095488 

22.1810730 

.8944463 

.002032520 

493 

243049 

119823157 

22.2036033 

.8997917 

.00.2028398 

494 

244036 

120553784 

22.2261108 

7.9051294 

.002024291 

495 

245025 

121287375 

22.2485955 

7.9104599 

.002020202 

496 

246016 

122023936 

22.2710575 

7.9157832 

.002016129 

TABLE  XXIII.— SQUARES,  CUBES,  SQUARE  ROOTS. 


,' 
No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

497 

217009 

122763473 

22.2934968 

.9210994 

.002012072 

498 

248004 

123505992 

22.3159136 

.9264085 

.002008032 

409 

249001 

124251499 

22.3383079 

.9317104 

.002004006 

500 

250000 

125000000 

22.3606798 

.9370053 

.002000000 

591 

251001 

125751501 

22.3830293 

.9422*31 

.001996008 

5P3 

252  04 

126506008 

22.4053565 

.9475733 

.001992032 

503 

253009 

127263527 

22.4276615 

,9528477 

.001988072 

504 

254016 

128024064 

22.4499443 

.9581144 

.001984127 

505 

255025 

128787625 

22.4722051 

.9633743 

.001980198 

50(3 

256036 

129554216 

22.4944438 

.9686271 

.001976285 

507 

257049 

130323843 

22.5166605 

.9738731 

.001972:387 

508 

258064 

131096512 

22.5388553 

.9791122 

.001968504 

509 

259081 

131872229 

22.5610283 

.9843444 

.001964637 

510 

260100 

132651000 

22.5831796 

7.9895697 

.001960784 

511 

261121 

133432831 

22.6053091 

7.9947883 

.001956947 

512 

262144 

134217728 

22.6274170 

8.0000000 

.001953125 

513 

263169 

135005697 

22.6495033 

8.0052049 

.001949318 

514 

264196 

135796744 

22.6715681 

8.0104032 

.001945525 

515 

265225 

136590875 

22.6936114 

8.0155946 

.001941748 

516 

286256 

137388096 

22.7156334 

8.0207794 

.001937984 

517 

267289 

138188413 

22.7376340 

8.02oC574 

.001934236 

518 

268324 

138991832 

22.7596134 

8.0311287 

.001930502 

519 

269361 

139798359 

22.  7815715 

8.0362935 

.001926782 

520 

270400 

140608000 

22.8035085 

8.0414515 

.001923077 

521 

271441 

141420761 

22.8254244 

8.0466030 

.001919386 

522 

272484 

142236648 

22.8473193 

8.0517479 

.0019157'09 

523 

273529 

143055667 

22.8691933 

8.0568862 

.001912046 

524 

274576 

143877824 

22.8910463 

8.0620180 

.001908397 

525 

275625 

144703125 

22.9128785 

8.0671432 

.001904762 

526 

276676 

145531576 

22.9346899 

8.0722620 

.0019,11141 

527 

277729 

146363183 

22.9564806 

8.0773743 

.001897533 

528 

278784 

147197952 

22.9782506 

8.0824800 

.001893939 

529 

279841 

148035889 

23.0000000 

8.0875794 

.001890359 

530 

280900 

148877000 

23.0217289 

8.0926723 

.001886792 

531 

281961 

149721291 

23.0434372 

8.0977589 

.001883239 

532 

283024 

150568768 

23.0651252 

8.1028390 

.001879699 

533 

284089 

151419437 

23.0867'928 

8.1079128 

.001876173 

534 

285156 

152273304 

23.1084400 

8.1129803 

.001872659 

535 

286225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

287296 

153990656 

23.1516738 

8.1230962 

.001865672 

537 

288369 

154854153 

23.1732605 

8.1281447 

.001862197 

538 

289444 

155720872 

23.1948270 

8.1331870 

.001858736 

539 

290521 

156590819 

23.2163735 

8.1382230 

.001855288 

540 

291600 

157464000 

23.2379001 

8.1432529 

.001851852 

541 

292681 

158340421 

23.2594067 

8.1482765 

.001848429 

542 

293764 

159220088 

23.2808935 

8.1532939 

.001845018 

543 

294849 

16>103007 

23.3023604 

8.1583051 

.001841621 

544 

295936 

160989184 

23.3238070 

8.16:33102 

.001838.235 

545 

297025 

161878625 

23.3452-351 

8.1683092 

.001834862 

546 

208116 

162771336 

23.3666429 

8.1733020 

.001831502 

547 

299209 

163667323 

23.3880311 

8.1782888 

.001828154 

548 

300:304 

164566592 

23.4093998 

8.ia32695 

.001824818 

549 

301401 

165469149 

23.4307490 

8.1882441 

.001821494 

550 

302500 

iflKKoOOO 

23.4520788 

8.1932127 

.001818182 

551 

303601 

167284151 

23.4733892 

8.1981758 

.001814882 

552 

304704 

168196608 

23.4946802 

8.2031319 

.001811594 

553 

305809 

169112377 

23.5159520 

8.2030825 

.001.808318 

554 

306916 

170031464 

23..  5372046 

8.2130271 

.001805054 

r,55 

308025 

170953875 

23.5584380 

8.2179657 

.001801802 

556 

309136 

171879616 

23.5796522 

8.2228085 

.001798561 

557 

310249 

172808693 

23.6008474 

8.2278254 

.001795332 

558 

311364 

173741112 

23.6220236 

8.2327463 

.001792115 

323 


CUBE  ROOTS,  AND   RECIPROCALS. 


No. 

U;[uares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

659 

312481 

174676879 

23.6431808 

1 
8.2376614      .001788909" 

560 

313600    175616000 

23.6643191 

8.2425706 

.001785714 

561 

314721 

176558481 

23.6854386 

8.2474740 

.001782531 

563 

315844 

177504328 

23.7065392 

8.2523715 

.001779359 

563 

316969 

178453547 

23.7'276210 

8.2572633 

.001776199 

564 

318096 

179406144 

23.7486842 

8.2621492 

.001773050 

565 

319225 

180362125 

23.7697286 

8.2670294 

,001769912 

566 

320356 

181321496 

23.7907545 

8.2719039 

.001766784 

567 

321489 

182284263 

23.8117618 

8.2767726 

.001768668 

568 

322624 

183250432 

23.8327506 

8.2816855 

.001760563 

569 

323761 

1842.iOUUO 

2U.868&909 

8.2864928 

.001757469 

570  1  324900 

185193000 

23.8746728 

8.2913444 

.001754386 

571  !  326041 

186169411 

23.8956063 

8.2161903 

.001751313 

572 

327184 

187149248 

83.9165215 

8.3010304 

.001748252 

573 

328329 

188132517 

23.9374184 

8.2058651 

.001745201 

574    329476 

189119224 

23.9582971 

8.3106941 

.001742160 

575 

330025 

190109375 

23.9791576 

8.3155175 

.001739130 

576    831776 

191102976 

24.0000000 

8.3203353 

.001736111 

577 

332929 

192100033 

24.0208243 

8.3251475 

.001733102 

578 

334084 

193100552 

24.0416306 

8.3299542 

.00173010-1 

579 

335241 

194104539 

24.0624188 

8.3347553 

.001727116 

580 

336400 

195112000 

24.0831891 

8.3S95509 

.001724138 

581 

837561 

196122941 

24.1039416 

8.3443410 

.001721170 

582 

338724 

197137368 

24.1246762 

8.3491256 

.001718218 

583 

339889 

198155287 

24.1453929 

8.3539047 

.C017  15266 

584 

341056 

199176704 

24.1660919 

8.3586784 

.001712329 

585 

342225 

200201625 

24.1867732 

8.8634466 

.C01709402 

586 

343396 

201230056 

24.2074369 

8.3G82095 

,001706485 

587 

344569 

202262003 

24.2280829 

8.3729668 

.001703578 

588 

345744 

203297472 

24.2487113 

8.3777188 

.001700680 

589 

316921 

204336469 

24.2693222 

8.3824653 

.001697793 

590 

848100 

205379000 

24.2899156 

8.387'2CC5 

.001694915 

591 

349281 

206425071 

24.3104916 

8.3919423 

.001692047 

592 

350464 

207474688 

24.3310501 

8.2966729 

.001689189 

593  !  351649 

208527857 

24.3515913 

8.4013881 

.001686341 

594 

352836 

209584584 

24.3721152 

8.4061180 

.001683502 

595 

354025 

210644875 

24.3926218 

8.4108326 

.001680672 

596 

355216 

211708736 

24.4131112 

8.4155419 

.001677852 

597 

356409 

212776173 

24.4335834 

8.4JC024CO 

.001675042 

598 

3:>7604 

213847192 

24.4540S85 

8.4249448 

.001672241 

599 

358801 

214921799 

24.4744765 

8.4296383 

.001669449 

GOO 

360000 

216000000 

24.4948974 

8.4343267 

.001666667 

001 

361201 

217'081801 

24.5153013 

8.4390098 

.001668894 

602 

362404 

218167208 

24.5356883 

8.4486877 

.001661180 

603 

363609 

219256227 

24.5560583 

8.4483605 

.001658375 

604 

364816 

220348864 

24.5764115 

8.4530281 

.001C55629 

605 

366025 

221445125 

24.5967478 

8.457CG06 

.001652893 

606 

S67236 

222545016 

24.6170673 

8.4623479 

.001650165 

607 

368449 

223648543 

24.6373,00 

8.  4670001 

.001647446 

608 

369664 

224755712 

24.6576560 

8.4716471 

.001644737 

609 

370881 

225866529 

24.6779254 

8.4762892  * 

.001642036 

610 

372100 

226981000 

24.6981781     8.4809261 

.001639344 

Gil 

373321 

228099131 

24.7184142 

8.4855579 

.001636(561 

C12 

374544 

229220928 

24.7386.338 

8.4901848 

.001633987 

613 

375769 

230346397 

24.7588368 

8.4948065 

.001631321 

614 

376996 

231475544 

24.7790234 

8.4994233 

.001628664 

615 

378225 

232608375 

24.7991935 

8.5040350 

.061626016 

616 

379456 

233744896 

24.8193473 

8.5086417 

.001623377 

617 

380689 

234885113 

24.8394847 

8.5132435      .001  620746 

618 

3S1924 

236029032 

24.8596058 

8.5178403 

.001618123 

619 

383161 

237176659 

24.8797106 

8.5224321 

.001615509 

620 

384400 

238328000 

24.8997992 

8.5270189 

.001612903 

324 


TABLE  XXIII.-SQUARES,   CUBES,  SQUARE  ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Hoots. 

Cube  Roots. 

Reciprocals. 

621 

385641 

239483061 

24.9198716 

8.5316009 

.001610306 

622 

386884 

240641848 

24.93U9278 

8.5361780 

.001607717 

623 

388129 

241804367 

24.9599079 

8.5407501 

.001605136 

624 

389376 

242970624 

24.9799920 

8.5453173 

.001602564 

625 

390625 

244140625 

25.0000000     8.5498797 

.001600000 

626 

391876 

245314376 

25.0199920     8.  )544372 

.001597444 

627 

393129 

246491883 

25.0399681     8.5589899 

.001594896 

628 

394384 

247673152 

25.0599*82 

8.5635377 

.001592:357 

6.29 

395641 

248858189 

25.0798724 

8.5680807 

.001589825 

630 

396900 

250047000 

25.0998008 

8.5726189 

.001587302 

631 

398161 

251239591 

25.1197134 

8.5771523 

.0015847'86 

632 

399424 

252435968 

25.1396102 

8.5816809 

.001582278 

633 

4l"0689 

253636137 

25.1594913 

8.5862047 

.001579779 

634 

401956 

254840104 

25.1793566 

8.59072:38 

.001577287 

635 

403225 

256047875 

25.1992003 

8.5952380 

.001574803 

636 

404496 

257259456 

25.2190404 

8.5997476 

.001572327 

637 

405769 

258474853 

25.2388589 

8.6042525 

.001569859 

638 

407044 

259694072 

25.2586619 

8.6087526 

.001567398 

639 

408321 

260917119 

25.2784493 

8.6132480 

.001564945 

640 

409800 

262144000 

25.2982213 

8.6177388 

.001562500 

641 

410881 

283374721 

25.3179778 

8.6222248 

.001560062 

642 

412164 

264603288 

25.3377189 

8.6267:063 

.001557632 

643 

413449 

265847707 

25.3574447 

8.6311830 

.001555210 

644 

414736 

267089984 

25.3771551 

8.6356551 

.001552795 

645 

416025 

268335125 

25.3968502 

8.6401226 

.001550388 

646 

417316 

269586136 

25.4165301 

8.6445855 

.001547988 

647 

418609 

270840023 

25.4361947 

8.6490437 

.001545595 

648 

419904 

272097792 

25.4558441 

8.6534974 

.001543210 

649 

421201 

273359449 

25.4754784 

8.6579465 

.001540832 

650 

422500 

274625000 

25.4950976 

8.6623911 

.001538462 

651 

423801 

275894451 

25.5147016 

8.6668310 

.00153601)8 

652 

425104 

277167808 

25.5342907 

8.6712665 

.001533742 

653 

426409 

278445077 

25.5538647 

8.6756974 

.001531394 

654 

427716 

27972o2o4 

25.5734237 

8.6801237 

.001529052 

655 

429025 

281011375 

25.5929678 

8.6845456 

.001526718 

656 

430338 

282300416 

25.6124969 

8.68896:30 

.001524390 

657 

431649 

283593393 

25.6320112 

8.69*3759 

.001522070 

658  " 

432964 

284890312 

25.6515107 

8.6977843 

.001519757 

659 

434281 

286191179 

25.6709953 

8.7021882 

.001517451 

660 

435600 

287498000 

25.6904352 

8.7065877 

.001515152 

661 

436921 

288804781 

25.7099203 

8.7109827 

.001512859 

662 

438244 

290117528 

25.7293607 

8.71537'34 

.001510574 

663 

439569 

291434247 

25.7487864 

8.7197596 

.001508296 

664 

440893 

292754944 

25.7681975 

8.7241414 

.001506024 

665 

442225 

294079625 

25.7875939 

8.7285187 

.001503759 

666 

443556 

1895408296 

25.8069758 

8.7328918 

.001501502 

667 

444889 

296740963 

25.8263431 

8.737'2604 

.001499250 

668 

446224 

298077632 

25.8456960 

8.7'416246 

.001497'000 

669 

447561 

299418309 

25.8650343 

8.7459846 

.001494768 

670 

448900 

300763000 

25.8843582 

8.7503401 

.001492537 

671 

450241 

302111711 

25.9036677 

8.7546913 

.001490313 

672 

451584 

30:5464448 

25.9229628 

8.7590:383 

.001488095 

673 

452929 

304821217 

25.9422435 

8.7683809 

.001485884 

674 

454276 

306182024 

25.9615100 

8.7677192 

.001483680 

675 

4.55625 

307546875 

25.9807621 

8.7720532 

.001481481 

676 

456976 

308915776 

26.0000000 

8.776:3830 

.001479290 

677 

458329 

310288733 

26.0192237 

8.7807084 

.001477105 

678 

459684 

311665752 

26.0:384.331 

8.7850296 

.001474926 

679 

461041 

313046839 

£6.0576284 

8.7893466 

.004472754 

680 

462400 

314432000 

26.0768006 

8.7936593 

.001470.588 

681 

463761 

315821241 

26.0959767 

8.7979679 

.001468429 

682  |  465124 

317214568 

26.1151297 

8.8022721 

.001466276 

325 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

683 

466489 

318611987 

26.13426^7 

8.8065722 

.001464129 

68  1  I  467850 

320013504 

26.1533937 

8.8108681 

.001461988 

6*5 

469225 

321419125 

26.1725047 

8.8151598 

.001459854 

68(5 

470596 

322828850 

26.1916017 

8.8194474 

.001457720 

687 

471909 

324242703 

26.2100848 

8.8237307 

.001455604 

688 

473344 

325660072 

20.2297541 

8.8280099 

.001453488 

689 

474721 

327082769 

26.2488095 

8.8=322850 

.001451379 

690 

476100 

828509000 

26.2678511 

8.8365559 

.001449275 

691 

477481 

329939371 

20.2868789 

8.8408227 

.001447178 

692 

.478864 

33137I3888 

20.3058929 

8.8450854 

.001445087 

693 

480249 

332812557 

26.3248932 

8.8493440 

.001443001 

694 

481636 

334255384 

26.3438797 

8.8535985 

.001440922 

695 

483025 

335702375 

26.3628527 

8.8578489 

.001438849 

696 

484416 

&37153536 

26.3818119 

8.8620952 

.001436782 

697 

485809 

338608873 

26.4007576 

8.8663375 

.001434720 

698 

487204 

340008392 

26.4196896 

8.8705757 

.001432665 

699 

488601 

341532099 

26.4386081 

8.8748099 

.001430615 

700 

490000 

343000000 

26.4575131 

8.8790400 

.001428571 

701 

491401 

344472101 

26.4764046 

8.8832661 

.001426534 

703 

492804 

345948408 

26.4952820 

8.8874882 

.001424501 

703 

494209 

347428927 

26.5141472 

8.8917063 

001422475 

704 

495616 

348913664 

26.5329983 

8.8959204 

.001420455 

705 

497025 

350402625 

26.5518361 

8.9001304 

.001418440 

706 

498436 

351895816 

26.5706605 

8.9043366 

.001416431 

707 

499849 

353393243 

26.5894716 

8.9085:387 

.001414427 

708 

501204 

354894912 

26.6082094 

8.9127369 

001412429 

709 

502081 

356400829 

26  6270539 

8.9109311 

.001410437 

710 

504100 

357911000 

26.6458252 

8.9211214 

001408451 

711 

505521 

359425431 

26.6645833 

8.9253078 

.001406470 

712 

506944 

360944128 

26.6833281 

8.9294902 

.001404494 

713 

508369 

362467097 

26.7020598 

8.9336687 

.001402525 

714 

509796 

363994344 

26.7207784 

8.9378433 

.001400560 

715 

511225 

365525875 

20.7394839 

8.9420140 

.001398601 

716 

512656 

367061696 

26.7581763 

8.9461809 

.001396648 

717 

514089 

368601813 

26.7768557 

8.9503438 

001394700 

718 

515524 

370146232 

26.7955220 

8.9545029 

.001392758 

719 

516961 

371694959 

26.8141754 

8.9586581 

.001390821 

720 

518400 

373248000 

26.8328157 

8.9628095 

.001388889 

721 

519841 

374803361 

26.8514432 

8.9609570 

.(,01386963 

722 

521284 

376367048 

20.8700577 

8.  971  1007 

.001385042 

723 

522729 

377933067 

26.8886593 

8.9752406 

.001383126 

724 

524176 

379503424 

26.9072481 

8.9793766 

.001381215 

725 

525625 

381078125 

26.9258240 

8.9835089 

.001379310 

726 

527076 

382G57176 

26.9443872 

8.9876373 

.001377410 

727 

528529 

384240583 

26.9629375 

8.9917620 

.001375516 

723 

529984 

385828352 

26.9814751 

8.9958829 

.001373626 

729 

531441 

387420489 

27.0000000 

9.0000000 

.001371742 

730 

532900 

389017000 

27.01R5122 

9.0041134 

.001369863 

731 

534361 

390617891 

27.0370117 

9.0082229 

.001367989 

732 

535824 

392223168 

27.0554985 

9.0123288  ' 

.001366120 

733 

537289 

398833837 

27.0739727 

9.0164309 

.001364256 

734 

538756 

395446904 

27.0924344 

9.0205293 

.001362398 

735 

540*25 

397065375 

27.1108834 

9.0246239 

.001360544 

736 

541696 

398688256 

27.1293199 

9.0287149 

.00ia58696 

737 

543169 

400:315553 

27.1477439 

9.0328021 

.001356852 

738 

544644 

401947272 

27.1661554 

9.0368857 

.001355014 

739 

546121 

403583419 

27.1845544 

9.0409655 

.001353180 

740 

547600 

405224000 

27.2029410 

9.0450419 

.001351351 

741 

549081 

406869021 

27.2213152 

9.0491142 

.001349528 

742 

550564 

408518488 

27.2396769 

9.0531831 

.001347709 

743 

552049 

410172407 

27.2580263 

9.0572482 

.001345895 

744 

553536 

411830784 

27.2763634 

9.0613098 

.001344086 

TABLE  XXIII.— SQUARES,  CUBES,  SQUARE  ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

745 

555025 

413493625 

27.2946881 

9.0653677 

.001342282 

746 

556.116 

415160936 

27.3130006 

9.0694220 

.001340483 

747 

558009 

416832723 

27.3313007 

9.0734726 

.0013386^8 

748 

559504 

418508992 

27.3495887 

9.0775197 

.001330898 

749 

561001 

420189749 

27.3678644 

9.0815031 

.001335113 

750 

562500 

421875000 

27.3861279 

9.0a56030 

.001333333 

751 

564001 

423564751 

27.4043792 

9.  0890392 

.OM  331558 

752 

565504 

425259008 

27.4226184 

9.0930719 

.001329787 

753 

567009 

426957777 

27.4408455 

9.0977010 

.0013*8021 

754 

568516 

428661064 

27.4590604 

9.1017265 

.0013*6200 

755 

570025 

430368875 

27'.  4772633 

9.1057485 

.001324503 

756 

571536 

432081216 

27.4954542 

9.1097C69 

.001322751 

757 

573049 

433798093 

27.5136330 

9.1137818 

.001321004 

758 

574564 

435519512 

27.5317998 

9.1177931 

.001319261 

759 

576081 

437245479 

27'.  5499546 

9.1218010 

.001317523 

760 

577600 

438976000 

27.5680975 

9.1258053 

.001315789 

761 

579121 

440711081 

27.5862284 

9.  1*98)61 

.001314000 

762 

580844 

442450728 

27.0043475 

9.1338ft34 

.C01312336 

763 

5821  69 

444194947 

27.  62;  4546 

9.1377971 

.C0131U016 

764 

58-3696 

445943744 

27.6405499 

9.1417874 

.001308901 

765 

585225 

447697125 

27.6580334 

9.14577'42 

.001307190 

766 

586756 

449455096 

27.6767050 

9.1497576 

.001305483 

767 

588289 

451217663 

27.6947648 

9.1537375 

.001303781 

768 

589824 

452984832 

27.7128129 

9.1577189 

.G01S02083 

769 

591361 

454756009 

27.7308492 

9.1610869 

.001300390 

770 

592900 

456533000 

27.7488739 

9.1656565 

.001298701 

771 

594441 

458314011 

27.7068808 

9.1(JOG**5 

.001297017 

772 

595984 

460099648 

27.7848880 

9.1735852 

.001295337 

773 

597529 

461889917 

27.8028775 

9.1775445 

.001293001 

774 

599076 

4(53684824 

27.8208555 

9.1815003 

.001*919iiO 

775 

600625 

465484375 

27.8388218 

9.1854527 

.0012^0323 

776 

602176 

467288576 

27.8507766 

9.1894018 

.0012H8060 

777 

603729 

469097433 

27.8747197 

9.1933474 

.001287001 

778 

605284 

470910952 

27.8920514 

9.1972897 

.C01*h5247 

779 

606841 

472729139 

27.9105715 

9.2012286 

.€01283097 

780 

608400 

474552000 

27.9284801 

9.2051641 

.C01282051 

781 

609961  . 

476379541 

27.9463772 

9.20fJC902 

.0012h'0410 

782 

611524 

478211768 

27.9642029 

9.2120250 

.(i01*7'8772 

783 

613089 

480048687 

27.982137'2 

9.2109505 

.001277139 

784 

614656 

481890304 

28.0000000 

9.2208726 

.001275510 

785 

616225 

483736625 

28.0178515 

9.2247914 

.C01*7'3fc8o 

786 

617796 

485587656 

28.0356915 

9.2287008 

.001272*65 

787 

619369 

4874434C3 

28.0535203 

9.2326189 

.C01  270648 

788 

620944 

489303872 

28.0713377 

9.  2265277 

.001209036 

789 

622521 

491169069 

28.0891438 

8.24C4333 

.C01207427 

790 

624100 

493039000 

28.1069386 

9.2443355 

.001265823 

791 

625681 

494913671 

28.1247222 

9.2482344 

.€01264*23 

792 

627264 

496793088 

28.1424946 

9.2E21300 

.001202026 

793 

628849 

498677257 

28.1602557 

9.2EG0224 

.001201034 

794 

630436 

500566184 

28.1780056 

9.2599114 

.001*59446 

795 

632025 

502459875 

28.1957444 

9.21S7973 

.101257802 

796 

633616 

504358336 

28.2134720 

9.2676798 

.C0125C281 

797 

635209 

506261573 

28.2311884 

9.2715592 

.001*54705 

798 

636804 

508169592 

28.2488938 

9.27'54352 

.001253133 

799 

638401 

510082399 

28.2665881 

9.2793081 

.C01251504 

800 

640000 

512000000 

28.2842712 

9.2831777 

.001250000 

801 

641601 

513922401 

28.3019434 

9.2870440 

.001248439 

802 

G43204 

515849608 

28.3196045 

9.2909072 

.001246883 

803 

644809 

517781627 

28.8372546 

9.2947671 

.001245330 

804 

646416 

519718464 

28..  3548938 

9.2986239 

.0012437'81 

805 

648025 

521660125 

28.3725219 

9.3024775 

.001242230 

806 

649636 

523606616 

28.3901391 

9.3063278 

.001240695 

327 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

807 

651249 

525557943    28.4077454 

9.3101750    .001239157 

803 

6528(54 

527514112    28.4253408 

9.3140190 

.001237624 

809 

654481 

529475129 

28.4429253 

9.317a599 

.0012:36094 

810 

656100 

531441000 

28.4604989 

9.3216975 

.001234568 

811 

657721 

533411731 

28.4780617 

9.3255320 

.001233046 

812 

659344 

535387328 

28.4956137 

9.3293634 

.001231527 

813 

660969 

537367797 

28.5131549 

9.  3331916 

.001230012 

814 

662596 

539353144 

28.5306852 

9.3370167 

.001228501 

815 

684225 

541343375 

23.5482048 

9.3408386 

.001226994 

si  a 

665856 

543338490 

28.5657137 

9.3446575 

.001225490 

817 

667489 

545338513 

28.5832119 

9.3484731 

.001223990 

818     669124 

547343432 

28.6006993 

9.3522857 

.001222494 

819 

670761 

549353259 

28.6181760 

9.3:63352 

.001221001 

820 

672400 

551368000 

28.6356421 

9.3599016 

.001219512 

821 

674041 

553:387661 

28.6530976 

9.3637049 

.001218027 

822 

675684 

555412248 

28.6705424 

9.3675051 

.001216545 

823 

677329 

557441767 

28.6879766 

9.3713022 

.001215067 

824 

678976 

559476224 

28:7054098 

9.3750903 

.001213592 

825 

680625 

56151562.> 

28.7228132 

9.3788873 

.001212121 

826 

682276 

56:3559976 

28.7402157 

9.3828752 

.001210654 

827 

683929 

565609283 

28.7576077 

9.3864600 

.  0012091  W) 

828 

685584 

567863558 

28.77'49891 

9.3902419 

.001207729 

829 

687241 

569722789 

23.7923601 

9.3940206 

.001206273 

830 

688900 

571787000 

23.8097206 

9.3977964 

.001204819 

831 

690561 

573856191 

28.8270706 

9.4015691 

.001203369 

832 

692224 

575930368 

28.8444102 

9.4053387 

.001201923 

833 

693889 

578009537 

28.8617394 

9.4091054 

.001200480 

834 

695556 

580093704 

28.8790582 

9.4128690 

.001199041 

835 

697225 

582182875 

23.8963666 

9.4166297 

.001197605 

836 

698898 

584277056 

28.9136646 

9.4203873 

.001196172 

837 

700569 

586376253 

28.9309523 

9.4241420 

.001194743 

833 

702244 

588480472 

28.9482297 

9.4278936 

.001193817 

839 

703921 

590589719 

23.9654967 

9.4316423 

.001191895 

840 

705600 

592704000 

23.9827535 

9.4353880 

.001190476 

841 

707281 

594823321 

29.0000000 

9.4391307 

.001189061 

812 

708964 

596947688 

29.0172363 

9.4428704 

.001187648 

843 

710649 

599077107 

29.0344623 

9.4466072 

.001186240 

844 

712336 

601211584 

29.0516781 

9.4503410 

.001184834 

845 

714025 

603351125 

29.0688837 

9.4540719 

.001183432 

846 

715716 

605495736 

29.0860791 

9.4577999 

.001182033 

847 

717409 

607645423 

29.1032644 

9.4615249 

.001180638 

848 

719101 

609800192 

20.1204396 

9.4652470 

.001179245 

849 

720301 

611960049 

29.1376046 

9.4689661 

.001177856 

850 

722500 

614125000 

29.1547595 

9.4726824 

.001176471 

851 

724201 

616295051 

29.1719043 

9.4763957 

.001175088 

852 

725904 

618470208 

29.1890390 

9.4801061 

.001173709 

853 

727609 

620650477 

29.2061637 

9.4838136 

.001172333 

854 

729316 

622835864 

29.2232784 

9.4875182 

.001170900 

855 

731025 

625026375 

29.2403830 

9.4912200 

.001169591 

856 

732736 

627222016 

29.2574777 

9.4949188 

.001168224 

857 

734449 

629432793 

29.2745623 

9.4986147 

.001166861 

858 

73G1G4 

631628712 

29.2916370 

9.5023078 

.001165501 

859 

737831 

6338397,'9 

29.3087018 

9.5059980 

.001164144 

860 

739000 

636056000 

29.3257566 

9.5098a54 

.001162791 

831 

741321 

638277381 

29.3428015 

9.51:33699 

.001161-140 

8b2 

743344 

640503928 

29.3598365 

9.5170515 

.001160093 

883 

744769 

6427-35647 

29.3768616 

9.5207303 

.001158749 

864 

746196 

644972544 

29.3938769 

9.5244063 

.001157407 

8G5 

748225 

647214625 

29.4108823 

9.5280794 

.001156069 

806 

749956 

649461896 

29.4278779 

9.5317497 

.001154734 

807 

751689 

651714363 

29.4448637 

9.5354172 

.001153403 

868 

753424 

653972032 

29.4618397     9.5390818 

001152074 

328 


TABLE  XXIII.-SQUARES,  CUBES,  SQUARE  ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots.  Reciprocals. 

i         :  '•  '• 

869  . 

755161 

656234909 

29.4788059 

9.5427437 

.001150748 

870 

756900 

658503000 

29.4957624 

9.5464027 

.001149425 

871 

758641 

660776311 

29.5127091 

9.5500589 

.001148106 

872 

760:384 

G63054848 

29.5296461 

9.5537123 

.0011407*!) 

873 

762129 

665338617 

29.5465734 

9.5573630 

.001145475 

874 

763876 

667627624 

29.5634910 

9.5610108 

.001144105 

875 

765625 

66992187'5 

29.5803989 

9.5640559 

.001142857 

876 

767376 

672221376 

29.5972972 

9.5682982 

.001141553 

877 

769129 

674526133 

29.6141858 

9.5719377 

.001140251 

878 

770884 

67(5836152 

29*6310648 

9.5755745 

.001138952 

879 

772641 

679151439 

29.6479342 

9.5792085 

.001137050 

880 

774400 

681472000 

29.6647939 

9.5828397 

.001136364 

881 

776161 

683797841 

29.6816442 

9.5864082 

.00113.5074 

882 

777924 

686128968 

29.6984848 

9.5900939 

.001133787 

883 

779689 

688465387 

29.  7153159 

9.5937169 

.001132503 

884 

781456 

690807104 

29.7321375 

9.5973878 

.001131222 

885 

783225 

693154125 

29.7489496 

9.0009548 

.001129944 

886 

78-1996 

695506456 

29.7657521 

9.6045096 

.001128668 

887 

786769 

69:864103 

29.7825452 

9.6081817 

.001127396 

888 

788544 

700227072 

29.7993289 

9.6117911 

.001126126 

889 

790321 

702595369 

29.8161030 

9.6153977 

.001124859 

890 

792100 

704969000 

29.8328678 

9.6f90017 

.001123596 

891 

793881 

707347971 

29.8496231 

9.0220030 

.001122334 

892 

795664 

709732288 

29.8663(590 

9.0202016 

.001121076 

893 

797449 

712121957 

29.8831056 

9.6297975 

.001119821 

894 

799236 

714516984 

29.8998328 

9.6333907 

.0*31118568 

895 

801025 

716917375 

29.9105506 

9.  6309812 

.001117318 

896 

802816 

719323136 

29.9332591 

9.0405690 

.001116071 

897 

804609 

721734273 

29.9499583 

9.6441542 

.001114827 

898 

806404 

724150792 

29.9666481 

9.6477367 

.001113586 

899 

808201 

726572699 

29.9833287 

9.05131C6 

.001112347 

900 

810000 

729000000 

30.0000000 

9.6548938 

.001111111 

901 

811801 

731432701 

30.0166620 

9.0584684 

.001109878 

902 

813604 

733870808 

30.0333148 

9.6620403 

.001108047 

903 

815409 

736314327 

30.0499584 

9.6650096 

.001107420 

904 

817216 

738763264 

30.0665928 

9.6691762 

.0011115195 

905 

819025 

741217625 

30.0832179 

9.6727403 

.001104972 

906 

820836 

743677416 

30.0998339 

9.6763017 

.001103753 

907 

822649 

746142643 

30.1164407 

9.6798604 

.001102536 

908 

824464 

748613312 

30.1330383 

9.6834166 

.001101322 

909 

826281 

751089429 

30.1496269 

9.6869701 

.001100110 

910 

828100 

753571000 

30.1662063 

9.6905211 

.001098901 

911 

829921 

756058031 

30.1827765 

9.6940694 

.001097695 

912 

831744 

758550528 

30.1998377 

9.6976151  j  .001096491 

913 

833569 

761048497 

30.2158899 

9.7011583  j  .001095290 

914 

835396 

763551944 

30.  2324329 

9.7040989 

.001094092 

915 

837225 

766060875 

30.2489669 

9.7082369 

.001092896 

916 

839056 

768575296 

30.2654919 

9.7117723 

.001091703 

917 

840889 

771095213 

30.2820079 

9.7153051 

.001090r>13 

918 

842724 

-  773620632 

30.2985148 

9.7188354 

.001089&S 

919 

844561 

776151559 

30.3150128 

9.7223631 

.001088139 

020 

846400 

778688000 

30.3315018 

9.7258883 

.001086957 

921 

848241 

781229961 

30.3479818 

9.7294109 

.001085770 

922 

850084 

783777448 

30.3644529 

9.7329309 

.001084599 

923 

&51929 

786330467 

30.3809151 

9.73(54484 

.001083423 

924 

853776 

788889024 

30.3973683 

9.7399634 

.001082251 

925 

855625 

791453125 

30.4138127 

9.7434758 

.001081081 

926 

857476 

794022776 

30.4302481 

9.7469857 

.001079914 

927 

859329 

790597983 

30.4466747 

9.7504930 

.001078749 

928 

861184 

799178752 

30.4630924 

9.7539979 

.001077586 

929 

863041 

801765089 

30.4795013 

9.7575002 

.001076426 

930 

864900 

804357000 

30.4959014 

9.7610001 

.001075269 

329    . 


GUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Hoots. 

1 
Cube  Roots.  •  Reciprocals. 

931 

866761  |   806954491 

30.5122926 

9.7644974 

.001074114 

932 

868624     809557568 

30.52b67oO 

9.7679922 

.001072961 

933 

870489     812100237 

30.5450487 

9.7714845 

.001071811 

934 

872356     814780504 

30.5614136 

9.77'49743 

.001070(564 

935 

874*25 

817400375 

30.5777697 

9.7784616 

.001069519 

936 

876096 

820025856 

30.5941171 

9.7819466 

.001008376 

937 

877969 

822656953 

30.6104557 

9.7854288 

.001067236 

988 

879844     825293672 

30.6267857 

8.788908f 

.001066098 

939 

881721     827936019 

30.6431069 

9.792£861 

.001064963 

940 

883600    830584000 

30.6594194 

9.7958611 

.001068830 

941 

885481     833237621 

£0.6757233 

9.7998386 

.001062699 

942 

887364     835896888 

30.6920185 

9.8028036 

.001061571 

943 

889249 

8385(51807 

30.7083051 

9.8062711 

.001060445 

944 

891136 

841232384 

:;0.  7245830 

9.8097362 

.001059322 

945 

893025 

843908625 

30.7408523 

9.8131989 

.001058201 

94(5 

894916 

846590536 

30.7571130 

9.8166591 

.001057082 

947 

896809 

849278123 

30.7733651 

9.8201169 

.001055966 

948 

898704 

851971392 

30.7896086 

9.8235723 

.001054852 

949 

900601 

854670349 

30.80^8436 

9.8270252 

.001053741 

950 

902200 

857375000 

SO.  8220700 

9.8304757 

.001052632 

951 

904401 

860085351 

£0.8382879 

9.8339238 

.001051525 

952 

906304 

862801408 

30.8544972 

9.8373695 

.001050420 

953 

908209 

865523177 

30.8706981 

9.8408127 

.001049318 

954 

910116 

868250664 

80.8868904 

9.8442536 

.001048218 

955 

912025 

87098S875 

£0.i;C30743 

9.847CC20 

.001047120 

9(56 

913936 

873722816 

£0.9192497 

9.6511280 

.001046025 

957 

915849 

87(5467493 

30.9354166 

9.  i  545617 

.001044932 

958 

917764 

879217912 

£0.9515751 

9.8579929 

.001043841 

959 

919681 

881974079 

80.9677251 

9.6614218 

.001042753 

960 

921COO 

884736000 

80.9838C68 

9.8648483 

.00104.1667 

961 

923521 

687508681 

31.0COOCOO 

9.8682724 

.001040583 

962 

925444 

8S0277128 

31.0161248 

9.8716941 

.001089501 

963 

987868 

693056847 

31.0822413 

9.8751135 

.C01088422 

964 

929296 

895841344 

31.0483494 

9.8785305 

.001087344 

965 

931225 

8986S2125 

31.0644491 

9.8819451 

.C010S6269 

966 

933156 

901428696 

81.0805405 

9.8868574 

.C01C85197 

967 

935089 

9042310&3 

31.0966236     9.8887673 

.001(34126 

968 

937'024 

907039232 

31.1126984 

9.8921749 

.001058058 

969 

938961 

909853209 

31.1287648 

9.6955801 

.001031192 

970 

940900 

912673000 

31.1448230 

9.8£88830 

.001030928 

971 

942841 

915498611 

31.16C8729 

9.  IX  £8635 

.001029866 

972 

944784 

918330048 

31.1769145 

9.  £057  817 

.C01G26607 

973 

946729 

921167317 

31.19S9479 

9.9091776 

.001027749 

974 

948676 

924010424 

31.  £089731 

9.9125712 

.C01026C94 

975 

950625- 

926859375 

31.2249900 

9.915£624 

.001025641 

976 

952576 

929714176 

31.24C9987 

9.9198513 

.001024590 

977 

954529 

932574&S3 

31.2569992 

9.9227379 

.C01023541 

978 

956484 

9a5441352 

31.2729915 

9.S261222 

.0010S2495 

979 

958441 

938313739 

31.2889757 

9.9295042 

.001021450 

980 

960400 

941192000 

31.3049517 

9.9328889* 

.C01020408 

981 

988861 

944076141 

31.82C9195 

9.  £86261  3 

.00101t'868 

982 

964324 

946966168 

31.83C8792 

9.9896863 

.001016380 

983 

966289 

949862087 

31.3528808 

9.9430092 

.001017294 

984 

968256 

952763904 

31.3687743 

9.9463797 

.  00101  C260 

985 

970225 

955671625 

31.3847097 

9.9497479 

,001015228 

986 

972196 

958585256 

31.4006369 

9.9531188 

.001014199 

987 

974169 

961504803 

31.4165561 

9.95(54775 

.001013171 

988 

976144 

964430272 

31.4824673 

9.9598389 

.001012146 

989 

978121 

967361669 

31.4483704 

9.9631981 

.001011122 

9PO 

980100 

970299000 

31.4642654 

9.S665549 

.001010101 

S91 

982081 

973242271 

31.4801525 

9.9699095 

.001009082 

992 

984064 

976191488 

31.4960315 

9.9732619 

.001008065 

330 


TABLE  XXIII.— SQUARES,  CUBES,  ETC. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

903 

98G049 

979146657 

31.5119025 

9.9766120 

.001007049 

994 

988038 

982107784 

31.5277655 

9.9799599 

.001006036 

995 

990025 

985074875 

31.5436206 

9.9833055 

.001005025 

996 

992016 

988047936 

31.5594677 

9.9866488 

.001004016 

997 

9*1009 

991026973 

31.5753068 

9.9899900 

.001003009 

998 

936004 

994011093 

31.5911380 

9.99=33289  I  .001002004 

999 

998001 

997002999  1  31.6069613 

9.9966656 

.001001001 

1000 

1000000 

1000000000    31.6237766 

10.0000000 

.001000000 

1001 

1002001 

1003003001 

31.6:385840 

10.00:33322 

.0009990010 

100-2 

1034004 

10060120J3 

31.6543836 

10.0006623 

.0009980040 

1003 

1005009 

1009027027 

31.6701752 

10.0099899 

.  0009970090 

1004 

1008016 

1012  43064 

31.6859590 

10.0133155 

.0009960159 

1005 

1010035 

1015075125  1  31.7017349 

10.0166389 

.0009950249 

1006 

1012036 

1018103216  I  31.7175030 

10.0199601 

.0009940358 

1007 

1014049 

1021147343  !  31.7332633 

10.0332791 

.0009930487 

1003 

1016084 

1024192512  !  31.7490157 

10.0265958 

.00099206:35 

1009 

1018031 

1027243739  i  31.7647603 

10.0299104 

.0009910803 

1010 

1030100 

1033301000  i  31.7804972 

10.0332228 

.0009900990 

1011 

1032121 

1033364331  j  31.7963362 

10.0365330 

.0^,09891197 

1013 

1034144 

1038433723    31.8119474 

10.0398410 

.0009881423 

1013 

1026169 

1039509197  |  31.8276609 

10.0431469 

.0009871668 

1011 

1038195 

1042590744    31.8433666 

10.0464506 

.0009861933 

1015 

1030235 

1045678375 

31.8590646 

10.0497521 

.0009852217 

1016 

1032256 

1048772096 

31.8747549 

10.0530514 

.0009842520 

1017 

1034389 

1051871913 

31.8904374 

10.0563485 

.0009832842 

1018 

1035324 

1054977833 

31.9081123 

10.0596435 

.0009823183 

1019 

1038361 

1053039859 

31.9217794 

10.0629364 

.0009818543 

10.30 

1040400 

1031208000 

31.9374388 

10.0662271 

.0009803922 

1021 

1042441 

1064332261 

31.9530906 

10.0695156 

.0009794319 

1022  . 

1044484 

1087462648 

31.9637347 

10.0723020 

.0009784736 

1023 

1046529 

10r0599167 

31.9843712 

10.0760863 

.0009775171 

102  i 

1048576 

1073741824 

3.3.t>000030 

10.0793684 

.0009765625 

1033 

1050325 

1076890335 

33.0153212 

10.0823484 

.0009756098 

1023 

1052376 

1030045576 

32.0312348 

10.0359203 

.0009746589 

1027 

1054729 

10333)8833 

32.0463407 

10.0392019 

.0009737098 

1028 

1056784 

1036373953 

32.0834391 

10.0924755 

.0009727626 

1029 

1058841 

1039547339 

33.0780293 

10.0957469  . 

.0009718173 

1030 

1030900 

1092  727000 

32.0933131 

10.0990163 

.0009708738 

1031 

1052961 

1095912791 

32.1091887 

10.1023835 

.0009699321 

1032 

1035024 

1099104763 

32.1347563 

10.1055487 

.00  9689922 

1033 

1067039 

1102302937 

32.1403173 

10.1088117 

.0009680542 

1031 

1069156 

1105507304 

32.1558704 

10.1120726 

.0009671180 

1035 

1071225 

1103717875    32.1714159 

10.1153314 

.000960  IHSli 

1036 

1073296 

1111934656    32.18(59539 

10.1185882 

.0009(552510 

1037 

1075389 

1115157653    32.2024844 

10.1218428 

.0009643202 

1038 

1077444 

1118338372    32.2180074 

10.1250953 

.0009633911 

1039 

1079521 

1121622319 

32.2335229 

10.1283457 

.0009634639 

1010 

1031600 

1124884000 

32.2490310 

10.1315941 

.0009615385 

1041 

1033631 

1128111931 

32:2645316 

10.1348403 

.0009606148 

1012 

1035734 

1131333038 

32.2800248 

10.1380845 

.0009596929 

1043 

1037849 

1134626507 

32.2955105 

10.1413266 

.0009587738 

1044 

1039936 

1137893184 

32.3109883 

10.1445667 

.0009578544 

1045 

1093035 

1141166125 

32.3264598 

10.1478047 

.0039569378 

1046 

1094116 

1144445336 

32.3419233 

10.1510406 

.Ot309560229 

1047 

1093209 

1147730823 

33.3573794 

10.1542744 

.0009551098 

1048 

1093304 

1151022592 

83.  3728-381 

10.1575062 

.0009541985 

1049 

1100401 

1154330848 

32.3882695 

10.1607-359 

.0009532888 

1050 

1102500 

1157625000 

32.4037035 

10.1639636 

.0009523810 

1051 

1104601 

1160935651 

32.4191301 

10.1671893 

.0009514748 

1053 

1108704 

1164252608 

32.434.5495 

10.1704129 

.0009505703 

1053 

1108809 

1167575877 

32.449D615 

10.1736344 

.0009496676 

1054 

1110916 

1170905464    32.4653662 

10.1768539 

.0009487666 

331 


TABLE  XXIV.- LOGARITHMS  OF  NUMBERS. 


No. 

100  L.  000.] 

[No.  109  L.  040. 

N. 

0 

1 

2 

8         4 

5 

6 

7 

8          9 

Diff. 

100 

000000     0434 

0868 

1301      1734 

2166 

2598 

3029 

3461     3891 

432 

1 

4321  1  4751 

5181 

5609     6038 

6466 

6894 

7321 

7748     8174 

428 

* 

8600     QftOfi 

9451 

9876 

0300 

0724      1147 

1570 

1993     2415- 

. 

3 

012837     3259 

3680 

4100     4521 

4940     5360 

JOlU 

5779 

6197     6616 

420 

4" 

7033 

VAX.! 

7868 

8284     8700  i    9116      9532 

0361      0775 

416 

5 

021189 

1603 

2016 

2428     2841    j  3252 

3664 

4075 

4486     4896 

412 

6 

5306 

5715 

6125 

6533     6942    |  7350 

7757 

8164 

8571     8978 

408 

9384 

9789 

• 

0195 

0600      1004 

1408 

1812 

2216 

2619     3021 

404 

8 

033424 

3826 

4227 

4628     5029 

5430 

5*30 

6230 

6629     7028 

400 

7426 

7825 

8223 

8620      9017 

9414 

9311 

04 

0207 

0602     0998 

397 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

434 

43.4 

86.8 

130.2 

173.6 

217.0 

260.4 

3( 

>3.8 

347.2 

390.6 

433 

43.3 

86.6 

12 

9.9 

173.2 

216.5 

259 

8 

$ 

B.I 

346.4 

389.7 

432 

43.2 

86.4 

129.6 

172.8 

216.0 

259.2 

302.4 

345.6 

388.8 

431 

43.1 

86.2 

12 

9.3 

172.4 

215.5 

258 

6       3( 

H.7 

344.8 

387.9 

430 

|  43.0 

86.0 

12 

9.0 

172.0 

215.0 

258 

0       ft 

W.O 

344.0 

387.0 

429 

42.9 

85.8 

128.7 

171.6 

214.5 

257.4 

300.3 

343.2 

386.1 

428 

42.8 

85.6 

12 

8.4 

171.2 

214.0 

256 

8       21 

)!).6 

342.4 

385.2 

427 

42.7 

85.4 

128.1 

170.8 

213.5 

256 

2 

298.9 

341.6 

384.3 

426 

42.6 

85.2 

12 

7.8 

170.4 

213.0 

255 

6 

2< 

340.8 

383.4 

425 

42.5 

85.0 

127.5 

170.0 

212.5 

255 

0 

297.5 

340.0 

382.5 

424 

42.4 

84.8 

127.2 

169.6 

212.0 

254 

4 

2< 

)6.8 

339.2 

381.6 

423 

42.3 

84.6 

12 

6.9 

169.2 

211.5 

253 

8 

2< 

)6.1 

338.4 

380.7 

422 

42.2 

84.4 

126.6 

168.8 

211.0 

253 

2 

295.4 

337.6 

379.8 

421 

42.1 

84.2 

12 

6.3 

168.4 

210.5 

252 

6 

2< 

M.7 

336.8 

378.9 

420 

42.0 

84.0 

126.0 

168.0 

210.0 

252 

0 

294.0 

336.0 

378.0 

419 

41.9 

83.8 

IS 

5.7 

167.6 

209.5 

251 

4 

2< 

)3.3 

335.2 

377  1 

418 

1  41.8 

83.6 

IS 

5.4 

167.2 

209.0 

250 

a 

9 

)2.6 

334.4 

376!  2 

417 

41.7 

83.4 

125.1 

166.8 

208.5 

250.2 

a 

333.6 

375.3 

416 

!  41.6 

83.2 

IS 

4.8 

166.4 

208.0 

249 

i; 

2« 

)ll2 

332.8 

374.4 

415 

41.5 

83.0 

124.5 

166.0 

207.5       249.0 

290.5 

332.0 

373.5 

414 

41.4 

82.8 

124.2 

165.6       207.0  i     248 

4 

2! 

39.8 

331.2 

372.6 

413 

41.3 

82.6 

123.9 

165.2 

206.5  1     247 

.8 

» 

39.1 

330.4 

371.7 

412 

41.2 

82.4 

IS 

3.6 

164.8 

206.0 

247 

.2 

a 

38.4 

329.6 

370.8 

411 

41.1 

82.  S 

123.3 

164.4 

205.5       246 

.(i 

287.7 

328.8 

369.9 

410 

41.0 

82.  C 

IS 

3.0 

164.0 

205.0 

246 

.0 

21 

37.0 

328.0 

369.0 

40C 

40.9 

81.8 

IS 

2.7 

163.6 

204.5 

245 

.4 

21 

36.3 

327.2 

368.1 

408       40  8 

81.  € 

122.4 

163.2 

204.0 

244 

.8 

285.6 

326.4 

367.2 

40? 

40.7 

81.4 

li 

!2.1 

162.8 

203.5 

244 

.2 

21 

34.9 

325.6 

366.3 

406 

40.6 

81.  S 

121.8 

162.4 

203.0 

243 

6 

284.2 

324.8 

365.4 

405       40.5 

81.0 

121.5 

162.0 

202.5 

243.0 

21 

33.5 

324.0 

364.5 

404       40.4 

80.* 

! 

121.2 

161.6 

•202.0 

242.4 

a 

32.8 

323.2 

363.6 

40c 

, 

40.3 

80.  ( 

> 

V 

JO.  9 

161.2 

201.5 

241 

.8 

8 

32.1 

322.4 

362.7 

40$ 

• 

40.2 

80.' 

I 

V 

JO.  6 

160.8 

201.0 

241 

2 

S 

31.4 

321.6 

361.8 

401 

40.1 

80.  $ 

I 

V 

JO.  3 

160.4 

200.5 

240 

.0 

2 

BO.  7 

320.8 

360.9 

400' 

40.0 

80-0 

120.0 

160.0 

200.0 

240.0 

280.0 

320.0 

360.0 

39< 

) 

39.9 

79.  i 

J 

1 

19.7 

159.6 

199.5 

239 

.4 

2 

79.3 

319.2 

359.1 

3» 

} 

39.8 

79.1 

i 

119.4 

159.2 

199.0 

238.8 

278.6 

318.4 

358.2 

39' 

r 

39.7 

79.' 

1 

1 

19.1 

158.8 

198.5 

238 

.3 

2 

77.9 

317.6 

357.3 

396 

39.6 

79.2 

118.8 

158.4 

198.0 

237 

.6 

277.2 

316.8 

356.4 

39 

5       39.5 

79.0         118.5       158.0       197.5       237 

.0       276.5       316  0     355.5 

332 


TABLE  XXIV.-LOGARITHMS  OF  NUMBERS. 


No.  110  L.  041.] 

[No.  119  L.  078. 

N.         0 

1          2 

8         4 

5 

6 

7          8 

9 

Diff. 

110     041393 

1787     2182 

2576     2969 

3362 

3755 

4148     4540 

4982       £C3 

1          5o^i 

5714      0105 

6495     6bfc5 

7275      76C4 

8053      8442      8880       IW) 

2         92  Id 

DOUG    1/1)1)3 

(Y^J-irt       (Y?(\(\ 

1  1  JV-l           1  J\^>i 

f  ro 

8     153078 

3403     3S40 

UooU       l/fOO 

4230      4613 

1  1  OO    i     1  •  .)oo 

4WU      53,8 

5700      0142 

IC24 

8b3 

4         1)905 

r*.K'(t        r'ti(i(i 

8046      8420 

bh05     9185 

D;")(>'~]      !)()4£ 

C°20       9r"° 

5     060698 

1075      1452 

1829      2206 

2582 

2958 

3333      3709 

4083 

376 

6         4458 

4832     5206 

5580     5953 

6326 

6699 

7071      7443 

7815 

373 

7         8186 

8557     8928 

9298      9668 

AAQQ 

nr-i*"f»          -i-i  AK 

-tK-tA 

Q»7  A 

8     071882 

2250      2617 

2985      3352 

UUoo 

3718 

4085 

Ul  i  D          1140 

4451      4810 

1O14 

5182 

otv 
366 

9         5547 

5912      6276 

6640      7004 

7368 

7731 

8094      8457 

8819 

363 

1 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

-7 

8 

9 

395 
394 

39.5 
39.4 

79.0 

78.8 

118.5 
118.2 

158.0 
157.6 

197.5 
197.0 

237 

236 

.0 
.4 

276.5 
275.8 

316.0 
315.2 

355.5 

354.6 

393 

39.3 

78.6 

11 

7.9 

157.2 

196.5 

235 

.S 

275.1 

314.4     353.7 

392 

39.2 

78.4 

117.6 

156.8 

196.0 

235.2 

274.4 

HJ.-J.6      352.8 

391 

39.1 

78.2 

11 

7.3 

156.4 

195.5 

234 

.6 

273.7 

312.8      351.9 

390 

39.0 

78.0 

117.0 

156.0 

195.0 

234.0 

273.0 

312.0      351.0 

389 

38.9 

77.8 

11 

6.7 

155.6 

194.5 

283 

.4 

272.3 

311.2      850.1 

388 

38.8 

77.6 

116.4 

155.2 

194.0 

232.8 

271.6 

310.4     349.2 

387 

38  .-7 

77.4 

11 

6.1 

154.8 

193.5 

232 

.2 

270.9 

309.6      3-48.8 

386 

38.6 

77.2 

115.8 

154.4 

193.0 

231 

270.2 

3C8.8 

847.4 

385 

38.5 

77.0 

115.5 

154.0 

192.5 

231 

'.Q 

209.5 

308..  0 

346.5 

384 

38.4 

76.8    ' 

115.2 

153.6 

192.0 

2S0.4 

268.8 

307.2  ''845.6 

383 

38.3 

76.6 

11 

4.9 

153.2 

191.5 

228 

.8 

208.1 

306.  4      344.7 

38.2 

76.4 

114.6 

152.8 

191.0 

229.2 

267.4 

305.6     343.8 

381 

38.1 

76.2 

11 

4.3 

152.4 

ISO.  5 

228 

.6 

266.7 

804.8     342.9 

380 

38.0 

76.0 

114.0 

152.0 

190.0 

228.0 

206.0 

£04.0      3J2.0 

379 

37.9 

75.8 

11 

3.7 

151.6 

189.5 

227 

.4 

205.3 

E03.2 

841.1 

378 

37.8 

75.6 

113.4 

151.2 

189.0 

226.8 

264.6 

£02.4 

340.2 

377 

37.7 

75.4 

11 

3.1 

150.8 

188.5 

221 

.18 

203.9 

301.6      839.3 

376 

37.6 

75.2 

112.8 

150.4 

188.0 

££5.6 

203.2 

300.8     838.4 

375 

37.5 

75.0 

112.5 

150.0 

187.5 

225.0 

202.5 

300.0 

337.5 

374 

37.4 

74.8 

112.2 

149.6 

187.0 

224.4 

201.8 

299.2 

386.6 

373 

37.3 

74.6 

11 

1.9 

149.2 

180.5 

Kl 

.s 

201  .  1 

288.4 

835.7 

372 

37.2 

74.4 

111.6 

148.8 

186.0 

223.2 

260.4 

297.6 

884.8 

371 

37.1 

74.2 

11 

1.3 

148.4 

185.5 

*£ 

.0 

259.7 

296.8 

383.9 

37'0 

37.0 

74.0 

111.0 

148.0 

185.0 

2£2.0 

259.0 

2<J6.0 

383.0 

369 

30.9 

73.8 

11 

0.7 

147.6 

184.5 

221 

.4 

258.3 

205.2 

883.1 

368 

36.8 

73.6 

110.4 

147.2 

184.0 

220.8 

257.6 

£94.4 

381.2 

367 

36.7 

73.4 

11 

0.1 

146.8 

183.5 

22C 

.2 

2fi0.9 

293.6 

380.3 

366 

36.6 

73.2 

109.8 

146.4 

183.0 

219.6 

256.2 

£92.8 

329.4 

S65       36.5 

73.0 

109.5 

146.0 

•182.5 

219.0 

255.7 

292.0 

328.5 

364 

36.4 

72.8 

109.2 

145.6 

182.0 

218.4 

254.8 

291.2 

327.6 

363 

36.3 

72.6 

1( 

)8.9 

145.2 

181.  S 

217 

.S 

254.1 

2S0.4 

326.7 

362 

36.2 

72.4 

108.6 

144.8 

181.0 

217.2 

253.4 

289.6 

3*3.8 

361 

36.1 

72.2 

1( 

)8.3 

144.4 

180.5 

21  fc 

.0 

252.7 

288.8 

324  .  9 

360 

36.0 

72.0 

108.0 

144.0 

180.0 

216.0 

252.0 

288.0 

324.0 

359 

35.9 

71.8 

1( 

)7.7 

143.6 

179.5 

215 

.4 

251.3 

287.2 

823.1 

358 

35.8 

71.6 

1( 

)7.4 

143.2 

179.0 

214 

8 

250.6 

286.4 

322.2 

357 

35.7 

71.4 

107.1 

142.8 

178.5 

214.2 

249.9 

285.6 

321.3 

356 

35.6 

71.2 

106.8 

142.4 

178.0 

213.6 

249.2 

284.8 

320.4 

333 


TABLE  XXIV. -LOGARITHMS  OF  NUMBERS. 


No. 

120  L.  079.] 

[No.  134  L.  130. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8          9 

Diff. 

120 

079181      9543 

9904 

! 

1 

0266 

0626 

0987     1347 

1707 

2067     2426  f      300 

1 

O.S2785 

3144 

3503 

&861 

4219 

4576     4934 

5291 

5647     6004 

357 

2 

(33(30 

6716      7071 

7426 

7781 

8136 

8490 

8845 

9198     9552 

355 

(1(1  )-\ 

0258     0611 

0963 

1315 

1667 

2018 

237'0 

2721  i  3071 

352 

4 

0034*3 

3772 

412J 

4471 

4820 

5169 

5518 

5866 

6215     6.31,2 

349 

5 

6910 

7257 

7601 

7951 

8298 

8644 

8990 

9:335 

9681    

00°6 

346 

6 

100371 

0715      1059 

1403 

'1747 

'2091 

2434 

2777 

3119     3462 

343 

7 

3804 

4146 

4187 

4828 

5169 

i  5510 

5851 

6191 

6531      6871 

341 

8 

7210 

7549 

rsss 

8227 

8565 

8903 

9241 

9579 

9916    

0°53 

338 

9 

110590 

092(3 

1263 

1599 

1934 

2270 

2605 

2940 

3275     3609 

335 

130 

3943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608     6940 

333 

-i 

7271 

7603 

7(fil 

8265 

8595 

8926 

025fi 

9586 

9915 

'  V 

330 

2 

120574 

0903      1231 

1560 

1888 

2216 

2544 

2871 

3198     3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456      6781 

325 

4 
* 

7105 

7429 

7753 

8076 

8399 

!  8722 

9045 

9368 

9690 

13 

0012         323 

PROPORTIONAL  PARTS. 

Kft 

1 

2 

3 

4 

5 

6 

7 

8 

9 

355 

35.5 

71.0 

106.5 

142.0 

177.5 

213.0 

248.5 

284.0 

319.5 

854 

35.4 

70.8 

106.2 

141.6 

177.0 

212.4 

247.8 

283.2 

318.6 

353 

70.6 

10: 

<) 

] 

41.2 

176.5 

211.8 

247.1 

282.4 

317.7 

352 

35^2 

70.4 

105.6 

140.8 

176.0 

211.2 

246.4 

281.6 

316.8 

351 

35.1 

70.2 

105 

.3 

] 

40.4 

175.5 

210.6" 

245.7 

280.8 

315.9 

350 

8j.O 

70.0 

10? 

.0 

: 

40.0 

175.0 

210.0 

245.0 

280.0 

315.0 

849 

34.9 

69.8 

104.7 

139.6 

174.5 

209.4 

244.3 

279.2 

314.1 

318 

34.8 

69.6 

104 

.1 

J 

39.2 

174.0 

208.8 

243.6 

278.4 

313.2 

317 

34.7 

(ill.  I 

104.1 

138.8 

173.5 

208.2 

242.9 

277.6 

312.3 

346 

34.6 

69.2 

103.8 

133.4 

173.0 

207.6 

242.2 

276.8 

311.4 

3  15 

34.5 

69.0 

103.5 

138.0 

172.5 

207.0 

241.5 

276.0 

310.5 

344 

34.4 

68.8 

103 

.fl 

1 

37.6 

172.0 

206.4 

240.8 

275.2 

309.6 

343 

34.3 

68.6 

102.9 

137.2 

171.5 

205.8 

240.1 

274.4 

308.7 

312 

31.2 

68.4 

105 

.1; 

1 

38.8 

171.0 

205.2 

239.4 

273.6 

307.8 

341 

34.1 

68.2 

10; 

;> 

1 

36.4 

170.5 

204.6 

238.7 

272.8 

300.9 

840 

34.0 

68.0 

102.0 

136.0 

170.0 

204.0 

238.0 

27'2.0 

306.0 

339 

33.9 

67.8 

101 

.7 

1 

35.6 

169.5 

203.4 

237.3 

271.2 

305.1 

338 

33.8 

67.6 

101 

.4 

135.2 

169.0 

202.8 

236.6 

270.4 

304.2 

837 

33.7 

67.4 

101 

.1 

1 

34.8 

168.5 

202.2 

235.9 

269.6 

303.3 

'.',  ,ij 

33.  (3 

67.2 

100.8 

134.4 

168.0 

201.6 

235.2 

2C8.8 

302.4 

836 

83.5 

67.0 

100.5 

134.0 

167.5 

201.0 

234  5 

268.0 

301.5 

384 

OQ.    t 

66.8 

100 

.2 

1 

33.6 

167.0 

200.4 

233.8 

267.2 

300.6 

333 

88.8 

(3(3  (5 

99.9 

133.2 

166.5 

199.8 

233.1 

266.4 

299.7 

332 

33.2 

6(5!  4 

99 

.r, 

1 

32.8 

166.0 

199.2 

232.4 

265.6 

298.8 

331 

33.1 

66.2 

9G 

.3 

132.4 

165.5 

198.6 

231.7 

264.8 

297.9 

330 

33.0 

66.0 

9fl 

.0 

: 

32.0 

165.0 

198.0 

231.0 

264.0 

297.0 

829 

32.9 

65.8 

98 

.7 

31.6 

164.5 

197.4 

230.3 

263.2 

296.1 

328 

32.8 

65.6 

98.4 

131.2 

164.0 

196.8 

229.6 

262.4 

295.2 

327 

32.7 

65.4 

98 

1 

30.8 

163.5 

196.2 

228.9 

261.6 

294.3 

326 

32.6 

65.2 

97.8 

130.4 

163.0 

195.6 

228.2 

260.8 

293.4 

325 

32.5 

65.0 

97.5 

130.0 

162.5 

195.0 

227.5 

SfiO.O 

292.5 

321 

32.4 

64.8 

97.2 

129.6 

162.0 

194.4 

226.8 

259.2 

291.6 

323 

32.3 

64.6 

96 

.!) 

1 

29.2 

161.5 

193.8 

226.1 

258.4 

290.7 

322 

32.2 

64.4     !       96.6 

I 

28.8       161.0 

193.2 

225.4 

257.6 

289.8 

334 


TABLE    XXIV.  —  LOGARITHMS   OF  NUMBERS. 


No. 

ia5  L.  130.] 

[No.  149  L.  175. 

N. 

0 

1          2 

* 

45         C 

7 

8         9 

Diff. 

135 

130334     0655     0977 

1298 

1619    :  1939      2260 

2580 

2900  !  3219 

321 

6 

3539      3858     4177 

4496 

4814  I    5133  \  5451 

5769 

6086      64US 

318 

7 

6721      7037     7354 

7671 

7987 

;  8303  !  8618 

8934 

9249     9564 

316 

g 

9879 

0194     0508 

0822 

1136  !    1450 

1763 

2076 

2389     2702 

314 

q 

143015 

3327  |  3639 

3951 

4263 

•   4574     4885 

5196 

5507     5818 

311 

140 

6128 

6438     6748 

7058 

7367 

7676     7985 

8294 

8603     8911 

309 

9219 

9527  i  9835 

^ 

- 

. 

I   QT'Sfi       i/v*o 

1*5*70 

1  llTfi        1  0QO 

OA1? 

2 

152288 

2594     2900 

3205 

3510 

3815 

JLV/UU 

4120 

loiu 
4424 

1  U  i  0        1  Jo  w 

4728     5032 

OUrf 

305 

8 

5336 

5640     5943 

6246 

6549  i    6852 

7154 

7457 

7759     8061 

303 

A 

8362 

8664     89C5 

9567  1    9868 

0469 

0769      1068 

301 

5 

161368 

1667      1967 

2266 

2564 

2863 

3161 

3460 

3758     4055 

299 

6 

4358 

4650     4947 

5244 

5541 

5838 

6134 

6430 

6726      7022 

297 

"< 

7317 

7613     7908 

8203 

8497 

8792     9086 

9380 

9674     9968 

295 

8 

170262 

0555     0848 

1141 

1434 

1726 

2019 

2311 

2603     2895 

293 

9 

3186 

3478     3769 

4060 

4351 

4641 

4932 

5222 

5512     5802 

291 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

a-ai 

32.1 

64.2 

96.3 

128.4 

160.5 

192.6 

224.7 

256.8 

288.9 

320 

32.0 

64.0 

96.0 

128.0 

160.0 

192.0 

224.0 

256.0 

288.0 

319 

81.9 

63.8 

95 

7 

127.6 

159.5 

191 

4 

« 

3.3 

255.2  i  287.1 

318 

31.8 

63.6 

95 

4 

127.2 

159.0 

190 

8 

2S 

2.6 

251.4     286.2 

3  17 

31.7 

63.4 

95 

1 

126.8 

158.5 

190 

a 

221.9 

253.6     285.3 

310 

31.6 

63.2 

94 

8 

126.4 

158.0 

189.6 

221.2 

252.8     284.4 

815 

31.5 

63.0 

94* 

5 

126.0 

157.5 

189 

0 

2X 

0.5 

252.0 

283.5 

314 

31.4 

62.8 

94 

2 

125.6 

157.0 

188 

4 

21 

9.8 

251.2 

282.6 

313 

31.3 

62.6 

93.9 

125.2 

156.5 

187.8 

219.1 

250.4  '  281.7 

312 

31.2 

62.4 

93 

G 

124.8 

156.0 

187 

2 

218.4 

249.6 

280.8 

311 

31.1 

62.2 

93.3 

124.4 

155.5 

186 

6 

217.7 

248.8 

279.9 

310 

31.0 

62.0 

93.0 

124.0 

155.0 

186.0 

217.0 

248.0 

279.0 

809 

30.9 

61.8 

9i 

7 

123.6 

154.5 

185 

4 

21 

6.3 

247.2 

278.1 

308 

30.8 

61.6 

92.4 

123.2 

154.0 

184.8 

215.6 

246.4 

217.2 

307 

30.7 

61.4 

92 

1 

122.8 

153.5 

184 

8 

21 

4.9 

245  6 

276.3 

30G 

30.6 

61.2 

91 

8 

122.4 

153.0 

183.6 

214.2 

244.8 

275.4 

305 

30.5 

61.0 

91 

5 

122.0 

152.5 

183 

0 

21 

3.5 

244.0 

274  .5 

304 

30.4 

60.8 

91 

2 

121.6 

152.0 

182 

4 

212.8 

243.2 

273.6 

303 

30.3 

60.6 

90 

9 

121.2 

151.5 

181 

S 

21 

2.1 

242.4 

272.7 

302 

30.2 

60.4 

90.6 

120.8 

151.0 

181.2 

211.4 

241.6      271.8 

301 

30.1 

60.2 

90.3 

120.4 

150.5 

180.6 

210.7 

240.8     270.9 

300 

30.0 

60.0 

90 

0 

120.0 

150.0 

180 

0 

21 

0.0 

240.0     270.0 

299 

29.9 

59.8 

89 

f 

119.6 

149.5 

179 

4 

2C 

9.3 

239.2 

269.1 

293 

29.8 

59.6 

89.4 

119.2 

149.0 

178.8 

208.6 

.238.4 

268.2 

297 

29.7 

59.4 

89 

.1 

118.8 

148.5 

178 

a 

207.9 

237.6 

267.3 

296 

29.6 

59.2 

88 

.8 

118.4 

148.0 

177 

(i 

* 

>7.2 

236.8 

266.4 

295 

29.5 

59.0 

88 

5 

118.0 

147.5 

177 

0 

at 

>6.5 

236.0 

265.5 

294 

29.4 

53.8 

88.2 

117.6 

147.0 

176 

4 

2C 

5.8 

235.2 

264.6 

293 

29.3 

58.6 

87 

g 

117.2 

146.5 

175 

8 

205.1 

234.4 

263.7 

292 

29.2 

58.4 

87.6 

116.8 

146.0 

175 

$ 

24.4 

233.6 

262.8 

291 

29.1 

58.2 

87.3 

116.4 

145^5 

174 

(I 

203.7 

232.8 

261.9 

290 

29.0 

58.0 

87 

0 

116.0 

145.0 

174 

0 

2t 

3.0 

232.0 

261.0 

289 

28.9 

57.8 

86 

7 

115.6 

144.5 

173 

4 

2( 

2.3 

231.2 

260.1 

288 

28.8 

57.6 

86 

4 

115.2  , 

144.0 

172 

8 

2C 

1.6 

230.4 

259.2 

287 

28.7 

57.4 

86.1 

114.8 

143.5 

172 

2 

200.9 

229.6      258.3 

286 

28.6 

57.2 

85 

.8 

114.4 

143.0 

171 

6       200.2       228.8  !  257.4 

335 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  150  L.  176.1 

[No.  169  L.  230. 

N. 

0 

1 

2 

3 

«  !!  • 

0 

7 

8 

9 

Diff. 

150 

176091      6881      61,70 

6959 

7248  i    7536 

7825 

8113 

8401 

8689 

289 

9839 

89(7      J264      Jo52 

0126  !'  0413 

0699 

0986 

1272 

1558 

287 

2 

181844     2129 

2415 

2700 

2985       3270 

3555 

3839 

4123 

4407 

285 

3 

4691 

4975 

5259 

5542 

5825       6108 

6391 

6674 

6956 

:  7239 

283 

4" 

7521 

CSilW 

Kl  K  1 

8306 

8647  i  j  8928      Q*>f>(> 

Qzion 

9771 

5 

2846 

279 

190aS2     0612  i  0892 

1171 

1451    :  1730     2010 

2289 

2567 

6 

3125     3403  !  3681 

8860 

4237       4514 

4792 

5069 

5346 

5623 

278 

7 

5<KX>     617(5 
8057     8932 

6453 
9206 

6729 
9481 

7005 
9755 

7281 

7556     7832 

8107 

8382 

276 

0029  1  0303 

0577 

£                    , 

1  1  9  A 

MM 

9 

201397 

167'0 

1943 

2216 

2488 

2761     3033 

3305 

3577 

3848 

272 

160 

4120 

4391 

4663 

4934 

5204 

5475  i  5746 

6016 

6286 

6556 

271 

1 

6826 

7096 

7365 

7634 

7904' 

8173 

8441 

8710 

8979 

9247 

269 

2 

9515 

97'83 

0051 

0319 

0586 

0853 

1121 

1388 

1654 

1921 

267 

3 

"212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

4 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

6 

220108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

7 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7C30 

258 

9 

7'887 

8144 

8400 

8657 

8913  I    9170 

9426 

9682 

9938 

23 

0193 

256 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

285       28.5 

57.0 

85 

.5 

114.0 

142.5 

171.0 

199.5 

228.0 

256.5 

284    !  28.4 

56.8         85 

.2 

113.6 

142.0 

170.4 

198.8 

227.2     255.6 

283 

28.3 

56.6 

84 

.9 

113.2 

141.5 

169.8 

198.1 

226.4     254.7 

282 

28.2 

56.4 

84 

.6 

112.8 

141.0 

169.2 

197.4 

225.6 

253.8 

281     i  28.1 

56.2 

84 

.3 

112  4 

140.5  I     168.6 

196.7 

224.8     252.9 

280       28.0 

56.0 

84 

.0 

112.0 

140.0  1     168.0 

196.0  i 

224.0     252.0 

279 

27.9 

55.8         83 

.7 

111.6 

139.5       167.4 

195.3 

223.2     251.1 

278 

27.8 

55.6 

83 

.4 

111.2 

139.0       166.8 

194.6 

£22.4     250.2 

277 

27.7 

55.4         83 

.1 

110.8 

138.5       166.2 

193.9 

221.6 

249.3 

27'6 

27.6 

55.2 

82.8 

110.4 

138.0'      165.6 

193.2 

220.8 

248.4 

275 

27.5 

55.0 

82.5 

110.0 

137.5       165.0 

192.5 

220.0 

2-17.5 

274 

27.4 

54.8 

82 

.2 

109.6 

137.0       164.4 

191.8 

219.2 

246.6 

273 

27.3 

54.6 

81 

.9 

109.2 

136.5       163.8       191.1 

218.4 

245.7 

272 

27.2 

54.4 

81.6 

108.8 

136.0 

163.2       190.4  i 

217.6 

244.8 

271 

27.1 

54.2 

81 

.3 

108.4 

"  135.5       162.6       189.7 

216.8 

243.9 

270 

27.0 

54.0 

81.0 

108.0 

135.0 

162.0 

189.0 

216.0 

243.0 

269 

26.9 

53.8 

80 

.7 

107.6 

134.5 

161.4       188.3  ; 

215.2 

242.1 

268 

26.8 

53.6 

80 

.4 

107.2 

134.0       160.8       187.6 

214.4 

241.2 

207 

26.7 

53.4 

80.1 

106.8 

133.5       160.2  !     186.9  1 

213.6 

240.3 

266 

26.6 

53.2 

79.8 

106.4 

133.0        159.6 

186.2  ! 

212.8 

239.4 

385 

26.5 

53.0 

79.5 

106.0 

132.5       159.0 

185.5  ' 

212.0 

238.5 

204 

26.4 

52.8 

79 

.2 

105.6 

132.0       158.4 

184.8  \ 

211.2 

237.6 

203 

26.3 

52.6 

78 

.9 

105.2 

131.5 

157.8 

184.1 

210.4 

236.7 

202 

26.2 

52.4 

78.6 

104.8 

131.0 

157.2 

183.4  i 

209.6 

235.8 

201 

26.1 

52.2 

78 

.3 

104.4 

130.5       156.6 

182.7 

208.8 

234.9 

260 

26.0 

52.0 

78.0 

104.0 

130.0       156.0 

182.0 

208.0 

234.0 

259 

25.9 

51.8 

77 

.7 

103.6 

129.5 

155.4 

181.3  ! 

207.2 

233.1 

258 

25.8 

51.6 

77 

.4 

103.2 

129.0 

154.8 

180.6 

206.4 

232.2 

257 

25.7 

51.4 

77 

.1 

102.8 

128.5 

154.2 

179.9 

205.6 

231.3 

256 

25.6 

51.2 

76 

.8 

102.4 

128.0 

153.6 

179.2 

204.8 

230.4 

255 

25.5  !     51.0 

76.5 

102.0 

1^7.5       153.0       178.5  ; 

204.0 

229.5 

336 


TABLE  XXIV. -LOGARITHMS  OF  NUMBERS. 


No.  170  L.  230.] 

[No.  139  L.  270. 

N1 

8                    A             i-v»r*i 

. 

V          UllL. 

170 

230449 

0704     09JO 

1215 

1470 

!    1724 

1979 

223-1 

2488 

2742       255 

1 

2996 

8250 

8.304 

3757 

4011 

4^64 

4517 

4770 

5023 

5276       253 

2 

6528 

5781 

(5033 

62H5 

6537 

6789      7041 

7292 

7'544 

7795 

252 

ft04fi 

8297 

8548 

8799 

£049 

9299      9550 

9800 

oU^tO 

AAFCA 

AOAA   1      n,^n 

4 

240549  |  0799 

1048 

1297 

1546 

1795     2044 

2233 

UUtXJ 

2541 

UoUU 

2790 

f**J\) 

243 

5 

3038 

3286 

3534 

37t2 

4030 

4277 

4525 

477'2 

5019 

5266 

243 

G 

5513 

5759 

6006 

6252 

0499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

89o4 

9198 

9443 

9687 

9932 

0176 

24ft 

8 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

iC-k\) 

243 

9 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

241 

1 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

g 

9513 

9746 

9980 

0213 

0446 

0679 

0312 

1144 

1377 

1609 

233 

7 

271842 

2074 

2303 

2538 

2770 

3001 

3233 

•*  MS 

3464^ 

3696 

3927 

232 

8 

4158 

4389- 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

0 

6462 

6692 

6921 

7151 

7380 

7609 

7S:-iS 

8067 

8296 

8525 

229 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

•7 

8 

9 

255 

25.5 

51.0 

76.5 

102.0 

127.5 

153 

0 

178.5 

204.0 

229.5 

254 

25.4 

50.8 

76.2 

101.6 

127.0 

152 

4 

17 

7.8 

203.2 

228.6 

253 

25.3 

50.6 

75.9 

101.2 

126.5 

151 

8 

17 

7.1 

202.4 

227.7 

252 

25.2 

50.4 

75.6 

100.8 

126.0 

151 

2 

176.4 

201.6 

226.8 

251 

25.1 

50.2 

75.3 

100.4 

125.5 

150 

6 

17 

5.7 

200.8 

225.9 

250 

25  0 

50.0 

75.0 

100.0 

125.0 

150 

0 

175.0 

200.0 

225.0 

249 

24.9 

49.8 

74.7 

99.6 

124,5 

149 

4 

17 

4.3 

199.2 

224.1 

248 

24.8 

49.6 

74.4 

99.2 

124.0 

148 

8 

17 

3.6- 

198.4 

223.2 

247 

24.7 

49.4 

74.1 

98.8       123.5 

148.2 

17 

2.9 

197.6 

222.3 

246 

24.6 

49.2 

73.8 

98.4 

123.0 

147 

6 

17 

2.2 

196.8 

221.4 

245 

24.5 

49.0 

73.5 

98.0       122.5 

147.0 

171.5 

196.0 

220.5 

244 

24.4 

48.8 

73.2 

97.6       122.0 

146 

4 

170.8 

195.2 

219.6 

243 

24.3 

48.6 

72.9 

97.2       121.5 

145. 

S 

17 

0.1 

194.4 

218.7 

242 

24.2 

48.4 

72.6 

96.8 

121.0 

145.2 

169.4 

193.6 

217.8 

241 

24.1 

48.2 

72.3 

96.4 

120.5 

144. 

(5 

16 

8.7 

192.8 

216.9 

240 

24.0 

48.0 

72.0 

96.0 

120.0 

144. 

0 

168.0 

192.0 

216.0 

239 

23.9 

47.8 

71  .7 

95.6 

119.5 

143. 

4 

16 

7.3 

191.2 

215.1 

238 

23.8 

47.6 

71.4 

95.2 

119.0 

142. 

s 

10 

6.6 

190.4 

214.2 

237 

23.7 

47.4 

71.1 

94.8 

118.5 

142. 

3 

165.9 

189.6 

213.3 

236 

23.6 

47.2 

70.8 

94.4 

118.0 

141. 

(i 

16 

5.2 

188.8 

212.4 

235 

23.5 

47.0 

70.5 

94.0 

117.5 

141. 

0 

164.5 

188.0 

211.5 

234 

23.4 

46.8 

70.2 

93.6 

117.0 

140. 

4 

163.8 

187.2 

210.6 

233 

23.3 

46.6 

69.9 

93.2 

116.5 

139. 

8 

16 

3.1 

186.4 

209.7 

232 

23.2 

46.4 

69.6 

92.8 

116.0 

139. 

X! 

162.4 

185.6 

208.8 

231 

23.1 

46.2 

69.  3 

92.4 

115.5 

138. 

6 

16 

1.7 

184.8 

207.9 

230 

23.0 

46.0 

69.0 

92.0 

115.0 

138. 

0 

16 

1.0 

184.0 

207.0 

229 

22.9 

45.8 

68.7 

91.6 

114.5 

137.4 

160.3 

183.2     206.1 

228 

22.8 

45.6 

68.4 

91.2 

114.0 

136. 

6 

15 

9.6 

182.4 

205.2 

227 

22.7 

45.4 

68.1 

90.8 

113.5 

136. 

2 

158.9 

181.6 

204.3 

226 

22.6 

45.2 

67.8 

90.4        113.0 

135.6 

158  2 

180.8 

203.4 

337 


TABLE  XXIV. -LOGARITHMS  OF  NUMBERS. 


No.  190  L.  278.]                                                                                     [No.  214  L.  332. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

190 

278754 

8982 

9211 

9439 

9667 

9895 

0123 

0351 

0578 

0806 

228 

1 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2G22 

2849 

3075 

227 

2 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

3 

5557 

5i'82 

6007 

6232 

6456 

:  6681 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8(596 

!  8920 

9143 

9366 

9589 

9812 

223 

5 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

223 

6 

2256 

2478 

2699 

2920 

3141 

88JJ3 

3584 

3804 

4025 

4246 

221 

| 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

8 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

oor:Q 

(XV~l 

gown 

9507 

9725 

on  i« 

oooo 

y\ji  i 

if&OJ 

«jy-to 

0161 

0378 

0595 

0813 

218 

200 

801030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

1 

3196 

3412 

362S 

3844 

4059 

i  4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566  :  5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

3 

4. 

7496 
nijoA 

7710  i  7924 
0040  i 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

: 

WBU 

Jty±o 

0056 

0268 

0481 

0693 

0906 

1118 

1330 

1542 

212 

5      311751 

1966 

2177 

2889 

2600 

2812 

30243 

3234 

3445 

3656 

211 

G 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

7 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

0354 

0562 

07G9 

0977 

1184 

1391 

1598 

1805 

2012 

207 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

1 

4282     4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

8380 

8^«5l 

8787 

8991 

9194 

9398 

9601 

9805 

0008 

0211 

203 

4 

330414 

C617 

0819 

1022      1225 

1427 

1630 

1832 

2034 

2236 

202 

PROPORTIONAL  PARTS. 

Diff. 

i 

2               3 

4 

5 

6 

7 

8 

9 

225 

22.5 

45.0 

67.5 

90.0 

112.5 

135.0 

157.5 

180.0 

202.5 

224 

22.4 

44.8 

67.2 

89.6 

112.0 

134.4 

156.8 

179.2 

201.6 

223 

22.3 

44.6 

66.9 

89.2 

111.5 

133.8 

156.1 

178.4 

200.7 

222 

22  2 

44.4 

66.6 

88.8 

111.0 

133.2 

155.4 

177.6      199.8 

221 

22  .'l 

44.2 

68.3 

88.4 

110.5 

132.6 

154.7 

176.8  i  198.9 

220 

22.0 

44.0 

66.0 

H8.0 

110.0 

132.0 

154.0 

176.0 

198.0 

219 

21.9 

43.8 

65.7 

87.6 

109.5 

131.4 

153.3 

175.2 

197.1 

218 

21.8 

43.6 

65.4 

87.2 

109.0 

130.8 

152.6 

174.4 

196.2 

217 

21.7 

43.4 

65.1 

86.8 

108.5 

130.2 

151.9 

173.6 

195.3 

21G 

21.6 

43.2 

64.8 

86.4 

108.0 

129.6 

151.2 

172.8 

194.4 

215 

21.5 

43.0 

64.5 

86.0 

107.5 

129.0 

150.fi 

172.0 

193.5 

211 

21.4 

42.8 

64.2 

85.6 

107.0 

128.4 

149.8 

171.2 

192.6 

213 

21.3 

42.6 

63.9 

85.2 

106.5 

127.8 

149.1 

170.4 

191.7 

212 

21.2 

42.4 

63.6 

F4.8 

106.0 

127.2 

148.4 

169.6 

190.8 

211 

21.1 

42.2 

63.3 

84.4 

105.5 

126.6 

147.7 

168.8 

189.9 

210 

21.0 

42.0 

63.0 

84.0 

105.0 

126.0 

147.0 

168.0 

189.0 

209 

20.9 

41.8 

62.7 

83.6 

104.5 

125.4 

146.3 

167.2 

188.1 

208 

20.8 

41.6 

62.4 

83.2 

104.0 

124.8 

145.6 

166  4 

187.2 

207 

20.7 

41.4 

62.1 

82.8 

103.5 

124.2 

144.9 

165.6 

186.3 

20G 

20.6 

41.2 

61.8 

82.4 

103.0 

123.6 

144.2 

164.8' 

185.4 

205 

20.5 

41.0 

61.5 

82.0 

102.5 

123.0 

143.5 

164.0 

184.5 

204 

20.4 

40.8 

61.2 

81.6 

102.0 

122.4 

142.8 

163.2 

183.6 

203 

20.3 

40.6 

60.9 

81.2 

101.5 

121.8 

142.1 

162.4 

182.7. 

202 

20.2 

40.4 

60.6 

80.8 

101.0 

121.2 

141.4 

161.6 

181.8 

TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  215  L.  332.]                                                                                        [No.  239  L.  380. 

N' 

T^lff 

. 

w 

Dill. 

215 

332438 

2640 

2842 

8044 

3246 

3417 

3649 

3850 

4051 

4253 

2C2 

6 

4434 

4055 

4856 

50c7 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

6460 

6060 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

3 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

0047 

OMft 

1  OO 

9 

340444     0042 

0841 

1039 

1237 

1435 

1632 

1830 

UvHi 

2028 

UA.40 

2225 

1UJ 

158 

220 

2423     2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

197 

1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5(J62 

6157 

196 

2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666      9860 

{\t\KA 

4 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

UUt>± 

1989 

193 

5 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

C 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

6026 

6217 

6408 

6599 

6790 

6981 

717'2 

7363 

7554 

7744 

191 

8 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

g 

9835 

0025 

0215 

0404 

0593 

0783 

0972 

1161 

1350 

1539 

•JQQ 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

icy 

188 

1 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

2 

5488 

5675 

5862 

6049 

6236 

;  6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

i  8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

0143 

0328 

0513 

0698 

0883 

185 

5 

~371068 

1253~ 

1437~ 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

184 

G 

2912 

3096 

3280 

3464 

3647 

8831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

38 

0030 

181 

PROPORTIONAL  PARTS. 

Diflf. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

202 

201 

20.2 
20.1 

40.4 
40.2 

60.6 
60.3 

80.8 
80.4 

101.0 
100.5 

121.2 
120.6 

141.4 
140.7 

161.6 
160.8 

181.8 
180.9 

200 

20.0 

40.0 

60.0 

80.0 

100.0 

120.0 

140.0 

160.0 

180.0 

199 

19.9 

39.8 

59.7 

79.6 

99.5 

119.4 

139.3 

159.2  !  179.1 

198 

19.8 

39.6 

59.4 

79.2 

99.0 

118.8' 

138.6 

158.4 

178.2 

197 

19.7 

39.4 

59.1 

78.8 

98.5 

118.2 

137.9 

157.6 

177.3 

196 

19.6 

39.2 

58.8 

78.4 

98.0 

117.6 

137.2 

156.8 

176.4 

195 

19.5 

39.0 

58.5 

78.0 

97.5 

117  0 

136.5 

156.0 

175.5 

194 

19.4 

38.8 

58.2 

77.6 

97.0 

116.4 

135.8 

155.2 

174.6 

193 

19.3 

38.  G 

57.9 

77.2 

9G.5 

115.8 

135.1 

154.4 

173  7 

192 

19.2 

38.4 

57.6 

76.8 

96.0 

115.2 

134.4 

153.6 

172.8 

1  1 

19.1 

38.2 

57  3 

76.4 

95.5 

114.6 

133.7 

152.8 

171.9 

190 

19.0 

38.0 

57.0 

76.0 

95.0 

114.0 

ias.0 

152.0 

171.0 

189 

18.9 

37.8 

56.7 

75.  G 

94.5 

113.4 

132.3 

151.2      170.1 

188 

18.8 

37.6 

56.4 

75.2 

94.0 

112.8 

131.6 

150.4 

169.2 

187 

18.7 

37  4 

56.1 

74.8 

93  5 

112.2 

130.9 

149.6 

168.3 

186 

18.6 

37.2 

55.8 

74.4 

93^0 

111.6 

130.2 

148.8 

167.4 

185 

18.5 

37.0 

55.5 

74.0 

92.5 

111.0 

129.5 

148.0 

166.5 

184 

18.4 

36.8 

55.2 

73.6 

92.0 

110.4 

128.8 

147.2 

165.6 

183 

18.3 

36.6 

54.9 

73.2 

91.5 

109.8 

128.1 

146.4 

164.7 

182 

18.2 

36.4 

54.6 

72.8 

61.0 

109.2 

127.4 

145.6 

163.8 

181 

18.1 

36.2 

54.3 

72.4 

90.5 

108.6 

126.7 

144.8 

162.9 

•180 

18.0 

36.0 

54.0 

72.0 

90.0 

108.0 

126.0 

144.0 

162.0 

179 

17.9 

35.8 

53.7 

71.6 

89.5 

107.4 

125.3 

143.2 

161.1 

_* 

TABLE  XXIV. -LOGARITHMS  OF  NUMBERS. 


No.  240  L.  380.] 

[No.  269  L.  431. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Difl. 

240 

380211 

0392 

0573 

0754 

0934 

i  1115 

1296 

1476 

1656 

1837 

181 

1 

2017 

2197 

2377 

2557 

2737 

j  2917 

3097  |  3 

^77 

345 

5 

3636 

180 

2 

3815 

3995 

4174 

4353 

4533 

4712 

4891      5070 

5249 

5423 

179 

3 

5636 

5785 

5964 

6142 

6321 

6499 

6677     6 

356 

703 

1 

7212 

178 

4 

7390 

7568 

7746 

£924 

8101 

8279 

8456 

8 

634 

881 

1 

8989 

178 

9166 

9343 

9520 

9698 

9875 

1 

0051 

O998 

A 

•ins 

058 

j 

0759 

177 

6 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

U-*vu 

2169 

2345 

2521 

176 

y 

2697 

2873 

3048 

3224 

3400 

a575 

3751 

3 

926 

410 

1 

4277 

176 

8 

4152 

4627 

4802' 

4977 

5152 

5326 

5501 

5 

676 

585 

) 

6025 

175 

9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

7940     8114 
9674  i  9847 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9501 

173 

* 

0020 

0192 

0365 

0538 

0711 

A 

QQ-t 

10?: 

a      iooQ 

173 

2 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

ULX*J 

2605 

lUOu         j.x^u 

2777     2949 

172 

3 

3121 

3292 

3464 

3635 

.3807 

3978 

4149 

4 

320 

449 

.> 

4663 

171 

4 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6 

029 

619 

9     6370 

171 

5 

6540 

6710 

6881 

7051 

7221 

7491 

7561 

731 

7901 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

'  9087 

9257 

£ 

426  i  959 

5 

9764 

169 

*7 

9933 

0102 

0271 

0440 

0609 

i  0777 

0946 

1114      1283 

1451 

169 

8 

411620      1788 

1956 

2124 

2293 

!  2461 

2629 

2 

796 

296 

i 

3132 

168 

9 

3300     3467 

3635 

3803 

3970 

i  4137 

4305 

4472 

463 

y 

4806 

167 

260 

4973      5140 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

1 

6641 

6807 

6973 

7139 

7306 

7472 

7(538 

7 

804 

797 

.» 

8135 

166 

2 

8301     8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

3 

9956 

0121 

0286 

0451 

0616 

0781 

0945 

1110 

1275 

1439 

165 

4 

421604      1768 

1933 

2097 

2261 

2426 

2590 

2 

754 

291 

s 

3082 

164 

5 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4882 

5045 

5208 

saw 

5534 

5697 

5860 

6 

023 

618 

; 

6349 

163 

7 

6511 

6074 

GH36 

6999 

7161 

7324 

7486 

1 

648 

781 

i 

7973 

162 

8 

81%     8297 

8459 

8621 

8783 

i  8944 

9106 

! 

268 

9429 

9591 

162 

g 

9752  '  flQl  •* 

43 

0075 

0236 

0398 

0559 

0720     0881 

1042  !  1203 

161 

PROPORTIONAL  PARTS. 

Diflf.       1 

2               3 

4 

5 

6 

106.8 

7 

8 

9 

160.2 

178         7.8 

35.6         53 

4 

71.2 

89.0 

124.6 

142.4 

177         7.7 

35.4         53 

1 

70.8 

88.5 

106.2 

123.9 

141.6 

159.3 

176         7.6 

35.  2         52 

8 

70.4 

88.0 

105.6 

123.2 

140.8 

158.4 

175         7.5 

a5.0         52 

5 

70.0 

7.5 

105.0 

122.5 

140.0 

157.5 

174         7.4 

34.8         52 

2 

69.6 

87.0 

104.4 

12U3 

139.2 

156.6 

173       17.3 

34.6         51 

9 

69.2 

86.5 

103.8 

121.1 

133.4 

155.7 

172       17.2 

34.4         51 

0 

68.8 

86.0 

103.2 

120.4 

137.6 

154.8 

171        17.1 

34.2         51 

3 

68.4 

85.5 

102.6 

119.7 

136.8 

153.9 

170       17.0 

34.0         51.0 

68.0 

85.0 

102.0 

119.0 

136.0 

153.0 

169       16.9 

33.8         50.7 

67.6 

84.5 

101.4 

118.3 

135.2 

152.1 

168       16.8 

33.6         50 

4 

67.2 

84.0 

100.8 

117.6 

134  4 

151  2 

167       16.7 

33.4         50 

1 

66.8 

83.5 

100.2 

116.9 

133.6 

150.3 

166       16.6 

33.2         49 

8 

66.4 

83.0 

99.6 

116.2 

132.8 

149.4 

165       16.5 

33.0         49 

5 

66.0 

82.5 

99.0 

115.5 

132.0 

148  5 

164       16.4 

32.8         49.2 

65.6 

82.0 

98.4 

114.8 

131.2 

147.6 

163       16.3 

32.6         48 

0 

65.2 

81.5 

97.8 

114.1 

130.4 

146.7 

162       16.2 

32.4         48 

5 

64.8 

81.0 

97.2 

113.4 

129.6 

145  8 

161        16.1 

32.2         48 

3 

64.4 

80.5 

96.6 

112.7 

128.8 

144.9 

340 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  270  L.  431.]                                                                                      [No.  299  L.  476. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

270 

431364     1525     1685 

1846 

2007 

2167 

2328 

2488 

2649      2809 

161 

1         29G9 

8130  1  3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

2 

4.o69 

4729     4838     5048 

5207 

5367  ,  5526 

5685 

5844 

6004 

159 

3 

6163     6322 

6481      6640     6799 

6957     7116 

7275 

74:33 

755W 

159 

4 

7751 

7909 

8067     8226 

8384 

8542     8701 

8859 

9017 

9175 

158 

5 

9333 

9491 

9(>48     9806 

9964 

1T"~  —  " 



1  K.Q 

6 

440909 

1066 

1224 

1381      1538 

1695      1852 

2009 

2166 

2323 

loo 

157 

7 

2480 

2637 

2793 

2950 

3106 

3~>G3      3419 

3576 

37:32  i  3889 

157 

8 

4045 

4201 

4357 

4513 

4669 

4825  !  4981 

•  5137 

5293  1  5449 

156 

9 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

824'i 

8397 

8552 

155 

1 

8706 

8861 

9015 

9170     9324 

9478 

9633     9787 

9941 

2 

450249 

0403  1  0557     0711 

0865 

1018 

1172 

1326 

1479 

1633 

154 

3 

1786 

1940 

2093     2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

•5 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

152 

6 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

757'9 

7731 

152 

7 

7882 

80:33 

8184 

8336 

8487 

8638 

8789 

8940. 

9091 

9242 

151 

g 

9392 

9543 

9694 

9845 

9995 

0146 

0296 

f\AA" 

CfXfJ 

a 

1M 

9 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

UOU  i 

2098 

2248 

lol 

150 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

ISO 

1 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085     9233 

9380 

9527 

9675 

148 

g 

9822 

9969 

0116 

0263 

0410 

0557 

0704 

0851 

0998 

1145 

147 

6 

471292 

1438" 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4G53 

4799 

4944  I  5090 

5235 

5381 

5526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397  ;  6542 

6687 

6832 

6976 

145 

PROPORTIONAL  PARTS. 

Diff.       1 

2 

3 

4 

5 

6 

7 

8 

9 

161       16.1 

32.2 

48.3 

64.4 

80.5 

96.6 

112.7 

128.8 

144.9 

160    1  16.0 

32.0 

48.0 

64  0 

80.0 

96.0 

112.0 

128.0 

144.0 

159       15.9 

31.8 

47.7 

63.6 

79.5 

95.4 

111.3 

127.2 

143.1 

158       15.8 

31.6 

.   47.4 

63.2 

79.0 

94.8 

110.6 

126.4 

142.3 

157       15.7 

31.4 

47.1 

62.8 

78.5 

94.2 

109.9 

125.6 

141.3 

156       15.6 

31.2 

46.8 

62.4 

78.0 

93.6 

109.2 

124.8 

140.4 

155       15.5 

31.0 

46.5 

62.0 

77.5 

93.0 

108.5 

124.0 

139.5 

154       15.4 

30.8 

46.2 

61.6 

77.0 

92.4 

107.8 

123.2 

138.  G 

153       15.3 

30.6 

45.9 

61.2 

76.5 

91.8 

107.1 

122.4 

137.7 

152       15.2 

30.4 

45.6 

60.8 

76.0 

91.2 

106.4 

121.6 

136.8 

151       15.1 

30.2 

45.3 

60.4 

75.5 

90.6 

105.7 

120.8 

135.9 

150       15.0 

30.0 

45.0 

60.0 

75.0 

90.0 

105.0 

120.0 

135.0 

149       14.9 

29.8 

44.7 

59.6 

74.5 

89.4 

104.3 

119.2 

134.1 

148       14.8 

29.6 

44.4 

59.2 

74.0 

88.8 

103.6 

118.4 

138.8 

147       14.7 

29.4 

44.1 

58.8 

73.5 

88.2 

102.9 

117.6 

132.3 

146       14.6 

29.2 

43.8 

58.4 

73.0 

87.6 

102.2 

116.8 

131.4 

145       14.5 

29.0 

43.5 

58.0 

72.5 

87.0 

101.5 

116.0 

130.5 

144       14.4 

28.8 

43.2 

57.6 

72.0 

86.4 

100.8 

115.2 

129.6 

143       14.3 

28.6 

42.9 

57.2 

71.5 

85.8 

100.1 

114.4 

128.7 

142       14.2 

28.4 

42.6 

56.8 

71.0 

85  2 

99.4 

113.6 

127.8 

141       14.1 

28.2 

42.3 

56.4 

70.5 

84.6 

98.7 

112.8 

126.9 

140       14.0 

28.0 

42.0 

56.0 

70.0 

84.0 

98.0 

112.0 

126.0 

TABLE  XXIV.-LOQARITHMS  OF   NUMBERS. 


No.  300  L.  477.] 

[No.  339  L.  531. 

N. 

0 

1 

2 

i 

4 

5 

6 

7 

8 

9 

Diff. 

145 

144 

144 
143 
143 
142 
142 
141 
141 

140 

140 
139 
139 
139 
138 
188 

1S7 
137 
186 
136 

136 
135 
135 

134 
134 
133 
183 
183 
182 
182 

131 

131 
131 
130 
180 
29 
29 
29 

28 
28 

300 
1 

2 
3 
4 
5 
6 

8 
9 

310 
1 
2 
3 
4 
5 
6 

7 
8 
9 

320 
1 

3 

4 
5 
6 

8 
9 

aso 
1 

a 

4 
5 
6 

7 
8 

9 

477121 

850(3 

7266 
8711 

7411 
8855 

7555 
8999 

7700 
9143 

7844 

9287 

7989 
9431 

8133  ! 
957o 

8278   8422 
9719   9663 

480007 
1443 
2874 
4300 
5721 
7138 
8551 
9958 

0151 
1586 
3016 
4442 

5863 
7280 
8692 

0099 

1502 
2900 
4294 

5683 
7068 
8448 
9824 

0294 
172*) 
3159 
4585 
6005 
7421 
8833 

0438 
1872 
3302 
4727 
6147 
7563 
8974 

0582 
2016 
3445 
4869 
6289 
7704 
9114 

0725 
2159 

3587 
5011 
6430 
7845 
9255 

0869 
2302'  j 
3730  ! 
5153  j 
6572  ' 
7986 
9396 

1012 
2445 

S872 
5295 
0714 
6127 
6537 

1156 
2568 
4015 
5437 
6855 
6269 
£677 

1C81 

2481 
8876 
6267 
CC53 
6C35 
S412 

0765 
2154 
8518 
4678 

€284' 
7E66 
6984 

€277 
1616 
2C51 
4£82 
ECC9 
6C82 
8251 

9EC6 

0676 

2183 
8486 
4785 
€C81 
7372 
8t(0 
SC43 

1299 
27,31 
4157 
E579 
6897 
8410 
9818 

1222 

2621 

4015 
5406 
6791 
8173 

C5EO 

0922 
2291 
£655 
£014 

CS70 
7721 
£018 

0239 

1642 
3040 
4433 

5822 
7206 
8586 
9962 

0380 

1782 
3179 
4572 
5960 
7344 
8724 

"0099" 
1470 
2837 
4199 

5557 
6911 
8260 
9606 

0520 

1922 
3319 
4711 
6099 
7483 
8862 

0236 
1607 
2973 
4335 

5693 
7046 
8395 
9740 

0661 

2062 
3458 
4850 
6238 
7621 
8999 

~0374 
1744 
,  3109 
4471 

5828 
!  7181 
!  8530 
9874 

1215 
2E51 
3883 
|  5211 
6535 
7855 

9171 

0801 

2201 
3597 
4989 
6376 
7759 
9137 

0511 
1880 
3246 
4607 

5964 
7316 

8664 

0941 

2341 

3737 
5128 
6515 
78S7 
9275 

0648 
2017 
8882 
4743 

CCG9 
7451 
6799 

0143 
1482 
2818 
4149 
5476 
68CO 
8119 

9434 

491302 
2760 
4135 
5544 
6930 
8311 
9687 

501059 
2427 
3791 

5150 
6505 
7856 
9203 

510645 

1883 
3218 
4548 
5874 
7196 

8514 

9828 

521138 
2444 
..3746 
5045 
8388 
7630 
8917 

1196 
2564 
3927 

5286 
6640 
7991 
9337 

13:33 
2700 
40G3 

5421 
6776 
8126 
9471 

OOC9 
1349 
2684 
4016 
£344 
6(568 
7987 

S303 

0411 
17EO 
8C84 
4415 
5741 
7CC4 
6£82 

CC97 

1CC7 
2314 
£616 
4915 
(210 
7t01 
8788 

0679 

2017 

a35i 

4681 
6006 
7328 

8646 
9959 

1269 
2575 
3876 
5174 
64(59 
7759 
9045 

0813 
2151 
3484 
4813 
6139 
7460 

8777 

0947 
2284 
3617 
4946 
6271 
7592 

8909 

1081 
2418 
3750 
5079 
6403 
7724 

9040 

0090 
1400 
2705 
4006 
5304 
6598 
7888 
9174 

0221 
1530 

2835 
4136 
5434 
6727 

8016 
9:;02 

0333 
1661 
2966 
4266 
6EG3 
6F56 
8145 
9430 

0464 
1792 
3096 
4396 
5693 
6985 
8274 
9559 

0615 
1922 
3226 
4526 

6822 
7114 
8402 
9687 

0745 
2053 
3356 
4626 
£951 
7243 
8531 
€815 

C072 
1351 

530200  0328  1  0456 

0584 

0712 

0840  0968  |  IC96   1223 

PROPORTIONAL  PARTS. 

Diff.   1 

2 

J 

J 

4 

55.6 
55.2 
54.8 
54.4 
54.0 
53.6 
53.2 
52.8 
52.4 
52.0 
51.6 
51.2 
50.8 

5 

6 

83.4,. 
82.8 
82.2 
81.6 
81.0 
80.4 
79.8 
79.2 
78.6 
70.0 
77.4 
76.8 
76.2 

r. 

97.3 
96.6 
95.9 
95.2 
94.5 
93.8 
93.1 
92.4 
91.7 
91.0 
90.3 
89.6 
88.9 

8 

111.2 
110.4 
109.6 
108.8 
1C8.0 
107.2 
TC6.4 
105.6 
104.8 
104.0 
103.2 
102.4 
101.6 

9 

125.1 
124.2 
123.3 
122.4 
121.5 
120.6 
119.7 
118.8 
117.9 
117.0 
116.1 
115.2 
114.3 

139   13.9 
138   13.8 
137   13.7 
136   13.6 
135   13.5 
134   13.4 
133   13.3 
132   13.2 
131   13.1 
130   13.0 
129   12.9 
128   12.8 
127   12  7 

27.8 
27.6 
27.4 
27.2 
27.0 
26.8 
26.6 
26.4 
26.2 
26.0 
25.8 
25.6 
25.4 

41.7 

41.4 
41.1 
40.8 
40.5 
40.2 
39.9 
39.6 
39.3 
39.0 
38.7 
38.4 
38.1 

69.5 
69.0 
68.5 
68.0 
67.5 
67.0 
66.5 
66.0 
65.5 
65.0 
64.5 
64.0 
63.5 

!U9. 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  340  L.  531.]                                  [No.  379  L.  679. 

N. 

0 

1 

2 

8 

H!6 

6 

7 

8 

9 

Diff. 

340 
1 
2 
3 
4 
5 
6 

7 
8 
9 

350 
1 
2 
3 
4 

5 

6 

» 

8 
9 

360 
1 
2 
3 

4 
5 
6 

7 
8 
9 

370 
1 

2 
3 
4 
5 
6 
7 
8 
9 

531479 
2754 
4026 
5294 
6558 
7819 

gore 

1607 
288-i 
4153 
5421 
0085 
7945 
9202 

1734   1802 
3009  3136 
4280  4407 
5547  i  5074 
0811   6937 
8071  !  8197 
9327  9452 

1990  ii  2117  2245 
3204   3391  i  3518 
4534   4001   4787 
5800   5927  0053 
7003   7189  7315 
8322  ;  8448  8574 
9578   9703  9829 

2372 
3045 
4914 
61PO 
7441 
8699 
9954 

2500 
3772 
5041 
0300 
7507 
8825 

0079 
1330 
25  Jo 
3820 

5060 
0290 
7529 
8758 
9984 

2627 

3899 
5107 
6432 
7093 
8951 

0204 
1454 
2701 
3944 

5183 
6419 
7652 

8881 

128 
127 
127 
126 
120 
126 

125 
125 
125 
124 

124 
124 
123 
123 

123 
122 
122 
121 
121 
121 

120 
120 
120 

119 
119 
119 
119 

118 
118 
118 

117 

117 
117 
116 
116 
116 
115 
115 
115 
114 

540329 
1579 
2825 

4068 
5307 
6543 

7775 
9003 

0455 
1704 
2950 

4192 
5431 
6066 
7898 
9126 

0580 
1829 
3074 

4316 
5555 

6789 
8021 
9249 

0705 
1953 
3199 

4440 

5078 
0913 
8144 
9371 

0830   0955 
2078   2203 
3323   3447 

4564  1  4688 
5802  •  5925 
7036   7159 
8267   8389 
9494   9016 

1080 
2327 
3571 

4812 
0049 

7282 
8512 
9739 

1205 
2452 
3096 

4936 
6172 
7405 
8035 
9801 

0106 
1328 
2547 
3762 
4973 
6182 

7387 
8589 
9787 

550228 
1450 
2668 
3883 
5094 

6303 
7507 
8709 
9907 

561101 
2293 

3481 
4066 
5848 
7026 

8202 
9374 

0351 
1572 
2790 
4004 
5215 

6423 

7627 

8829 

0473 
1694 
2911 
4126 
5336 

6544 

7748 
8948 

0595 
1816 
3033 
4247 
5457 

6664 
7868 
9068 

0717 
1938 
3155 
4308 
5578 

6785 
7988 
9188 

0385 
1578 
2769 
3955 
5139 
6320 
7497 

8671 
C842 

1010 
2174 
3336 
4494 
5050 
6802 
7951 
9097 

0840 
2000 
3276 

4489 
5699 

6905 
8108 
9308 

0962 
2181 
3398 
4610 

5820 

7026 
8228 
9428 

1084 
2303 
3519 
4731 
594U 

7146 
8349 
9548 

1206 
2425 
3640 
4852 
6001 

7267 
8469 
9607 

0026 
12^1 
2412 
3000 
4784 
5966 
7144 

8319 
9491 

0146 
1340 
2531 

3718 
4903 
0084 
7262 

8436 
9608 

0205 
1459 
2050 
3837 
5021 
6202 
7379 

8554 

9725 

0504 
1698 
2887 
4074 
5257 
6437 
7614 

8788 
9959 

0024 
1817 
3006 
4192 
5376 
6555 
7732 

8905 

0743 
1936 
3125 
4311 
5494 
0073 
7849 

9023 

0863 
2055 
3244 
4429 
5612 
6791 
7967 

9140 

0982 
2174 
3302 
4548 
5730 
6909 
80&4 

9257 

0076 
1243 
2407 
3508 
4726 
5880 
7032 
8181 
9326 

0193 
1359 
2523 
3084 
4841 
5996 
7147 
8295 
9441 

0309 
1476 
.2639 
3800 
4957 
6111 
7262 
8410 
9555 

0420 
1592 
2755 
3915 
5072. 
6226 
7377 
8525 
9009 

570543 
1709 
2872 
4031 
5188 
6341 
7492 
8639 

0660 
1825 
2988 
4147 
5303 
6457 
7607 
8754 

0776 
1942 
3104 
4263 
5419 
6572 
7722 
8868 

0893 
2058 
3220 
4379 
5534 
6687 
7836 
8983 

1126 
2291 
;  3452 
|  4610 
5765 
6917 
!  8006 
9212 

PROPORTIONAL,  PARTS. 

Diff.   1 

234 

5 

078 

9 

123   12.8 
127   12.7 
126   12.6 
125   12.5 
124   12.4 
123   12.3 
122   12.2 
121   12.1 
120   12.0 
119   11.9 

25.6    38.4    51.2 
25.4    38.1    50.8 
25.2    37.8    50.4 
25.0    37.5    50.0 
24.8    37.2    49.6 
24.6    36.9    49.2 
24.4    36.6    48.8 
24.2    36.3    48.4 
24.0    36.0    48.0 
23.8    35.7    47.6 

64.0 
63.5 
63.0 
62.5 
62.0 
61.5 
61.0 
60.5 
60.0 
59.5 

76.0    89.6 
76.2    88.9 
75.6    88.2 
75.0    87.5 
74.4    86.8 
73.8    86.1 
73.2    85.4 
72.6    84.7 
72.0    84.0 
71.4    83.3 

102.4 
101.6 
100.8 
100.0 
99.2 
98.4 
97.0 
96.8 
96.0 
95.2 

115.2 
114.3 
113.4 
112.5 
111.6 
110.7 
109.8 
108.9 
108.0 
107.1 

343 


TABLE  XXIV.-LOGARITHMS  OF  NUMBERS. 


No.  380.  L.  579.] 


[No.  414  L.  617. 


N. 

0 

1 

2 

3 

4 

5 

0 

7 

8 

9 

Diff. 

380 

579784 

9898 

I 

0128 

0355 

"0469~ 

0583 

0697  |  0811 

114 

0012 

0241 

1 

580925 

1039 

1153  j  1267 

1381 

1495  1608  '•  1722  1836 

1950 

2063 

2177 

2291   2404 

2518 

2631  !  2745  2S58  2972 

3085 

3 

3199 

3312 

3 

426 

853 

) 

3652 

3765  i  3879  i  39 

92  4105 

4218 

4 

4331 

4444 

4 

557 

467 

9 

4783 

4896  i  5009  51 

22  5235 

5348 

113 

5 

5461 

5574 

5686 

5799 

5912 

6024  i  6137  6250 

63G2 

6475 

6 

6587 

6700 

6 

312 

692 

& 

7037 

7149  7262  73 

74 

7486 

7599 

7 

7711 

7823  7 

(35 

804 

r 

8160  1  8272  |  8384  84 

96 

8608 

8720 

112 

8 

8832 

8944 

9056 

9167 

9279 

i  9391  j  9503 

9615 

9726 

9838 

9 

99j° 

0061 

0173 

0284 

0396 

0507 

0619 

0730 

0842 

0953 

3°0 

591065 

1176 

1287 

1399 

1510 

1621  1732 

1843 

1955 

2066 

1 

2177 

2288 

2399 

2510 

2621 

2732  2843  2954 

3064 

3175 

111 

2 

3286 

3397 

3 

508 

361 

8 

3729 

3840  3950 

4( 

161 

4171 

4282 

3 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

527'6 

5386 

4 

5495 

5606 

5 

717 

582 

7 

5937 

i  6047 

6157  65 

G7 

6S77 

6487 

5 

6597 

6707 

t 

817 

692 

7 

7037. 

7146 

7256  7£ 

JOG 

747'6 

7586 

110 

6 

7G'J5 

7805 

914 

8024  8134 

8243 

8353  8462 

8572 

8681- 

,  7 

8791 

8900 

9009 

9119  |  9228 

9337 

9446  !  9556 

9665 

9774 

8 

9883 

9992 



_ 

ri.  

'  '  







109 

0101 

0210  (319 

i  0428  0537 

0646 

0755 

0864 

9 

^00973 

1082 

1191 

1299 

1408 

j  1517 

1625 

1734 

1843 

1951 

400 

2060 

2169 

2277 

238 

ft  2494 

2603 

2711 

2819 

2928 

3036 

1 

3144 

3253 

g 

361 

34(3 

9  3577  !  3686 

3794  I  Sf 

to-,' 

4010 

4118 

108 

2 

4326 

4334 

4442 

4550  4658 

4766 

4874 

4< 

)S2 

5089 

5197 

3 

5305 

5413 

5 

521 

562 

18  5736 

5844 

5951 

6( 

r,',) 

6166 

6274 

4 

6381 

G4H9 

6596 

6704  1  6811 

6919 

7026 

7133 

7241 

7348 

5 

7455 

7562 

7 

669 

777 

7  i  7884 

7991 

8098 

8; 

.'05 

8312 

8419 

107 

6 

8526 

8633 

8740 

8847  i  8954 

9061 

9167 

9274 

9381 

9488 

7 

QTI1 

r 

S(~N 

991 

A  1 

f 

°  

y  i  ui 

OUo 

0021 

0128 

0234 

0341 

0447 

0554 

8 

610660 

0767 

0873 

097 

0 

1086 

1192 

1298 

1405 

1511 

1617 

9 

1723 

1829 

1 

936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

27'84 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

1 

3842 

3947 

4 

053 

41£ 

9 

4264 

4370 

4475  4 

Wl 

4686 

4792 

2 

4897 

5003 

I 

>108 

52 

3 

5319 

5424 

5529  |  5 

134 

5740 

5845 

3 

5950 

6055 

6160 

6265 

6370 

6476 

6581   6686 

6790 

6895 

105 

4 

7000 

7105 

7210 

7315 

7420  I!  7525 

7629 

7734 

7839 

7943 

PROPORTIONAL  PARTS. 

Diff.  j  1 

2 

3 

4 

5 

6 

7 

8 

9 

118   11.8 

23.6 

35.4 

47.2 

59.0 

70.8 

82.6 

94.4 

106.2 

117   11.7 

23.4 

35.1 

46.8 

58.5 

70.2 

81.9 

93.6 

105.3 

116   11.6 

23.2 

34.8 

46.4 

58.0 

69.6 

81.2 

92.8 

104.4 

115   11.5 

23.0 

34.5 

46.0 

57.5 

69.0 

80.5- 

92.0 

103.5 

114   11.1 

22.8 

34.2 

45.6 

57.0 

68.4 

79.8 

91.2 

102.6 

113   11.3 

22^6 

33.9 

45.2 

56.5 

67.8 

79.1 

90.4 

101.7 

112   11.2 

22.4 

33.6 

44.8 

56.0 

67.2 

78.4 

89.6 

100.8 

111   11.1 

22.2 

33.3 

44.4 

55.5 

66.6 

77.7 

88.8 

99.9 

110   11.0 

22.0 

33.0 

44.0 

55.0 

66.0 

77.0 

88.0 

99.0 

109   10.9 

21.8 

32.7 

43.6 

54.5 

65.4 

76.3 

87.2 

98.1 

108   10.8 

21.6 

32.4 

43.2 

54.  0 

64.8 

75.6 

86.4 

97.2 

107   10.7 

21.4 

32.1 

42.8 

53.5 

64.2 

74.9 

85.6 

96.3 

106   10.6 

21.2 

31.8 

42.4 

53.0 

63.6 

74.2 

84.8    95.4 

105   10.5 

21.0 

31.5 

42.0 

52.5 

63.0 

73.5 

84.0    94.5 

105   10.5 

21.0 

31.5 

42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

104   10.4 

20.8 

31.2 

41.6 

52.0 

62.4 

72.8 

83.2 

93.6 

344. 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  415  L.  618.]                                    [No.  459  L.  662 

i 

• 

9   Diff. 

415 

618048 

8153  !  8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

6 

9093 

9198  i  9302 

9406 

9511 

9615 

9719 

9824 

9928 

7  C20136 

0240  :  0344 

0448  0552 

0656  0760 

0864  !  0968 

1072 

104 

8    1176 

1280  i  1384 

1488 

1592 

1695 

1799 

1903  2C07 

2110 

9 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

1 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004  5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

-6032 

6135 

6238 

3 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

70E8 

7161 

72(i3 

4 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082  8185  8287 

5 

8389 

8491 

8593 

8695 

8797 

8900  9002 

9104  C206  9308 

102 

0410 

9512 

9613 

9715 

9817 

9919  

*j*±i\j 

sinoi 

n-lOO 

noo/4  !  /v-joA 

7 

630428 

0530 

0631 

0733 

0835  1  0926 

VUV.1 

10b8 

Ul^tJ 

1139 

\j&vt   Ud^O 

1241  i  1342 

8 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153  :  2255 

2356 

9 

2457 

2559 

2660 

2761 

2862 

2963 

8064 

3165  3266 

3367 

430 

3468 

3569 

3670 

3771 

3872 

•  3973  4074 

4175  4276  4376 

101 

1 

•4477 

4578 

4679 

4779 

4880 

i  4981  i  5081 

5182 

5^83  5S83 

2 

5484 

5584 

5685 

5785 

5886 

5986  i  6087 

6187 

6287  6388 

i  r  « 

3 

6488 

6588 

6688 

6789 

6889 

6989  7089 

718Q 

7290 

7390 

4 

7490 

7590 

7690 

7790 

7890 

7990  8090 

8190 

8290 

8389 

inn 

5 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387   -~ 

6 

9486 

9586 

9686 

9785 

9885 

9984 

















0084 

0183 

0283 

0382 

7 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

8 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

9 

2465 

2563 

2662 

2761 

2b60 

2959 

8058 

3156 

3255 

3354 

99 

440 

3453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

1 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

2 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

3 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

4 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

5 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919 



0016 

0113 

0210 

rr 

650308 

0405 

0502 

0599 

0696 

0793  0890 

0987 

1084 

1181 

8 

1278 

1375 

1472 

1569 

1666 

1762  1859 

1956 

2053 

2150 

87 

9 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

8019 

3116 

450 

3213 

3309 

3405 

3502 

3598 

|  3695 

3791 

3888 

3984  !  4080 

1 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

4850  :  4946  j  5042 

2 

5138 

5235 

5331 

5427 

5523 

;  5619  j  5715 

5810  !  5906  6002 

96 

3 

6098 

6194 

6290 

6482 

6577 

6673 

6769  6864  6960 

4 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725  i  7820  1  7916 

5 

8011 

8107 

8202 

8488  8584 

8679  8774  J  8870 

6 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631  j  9726  9821 

. 

"16 

0011 

0106 

0201 

0296 

1  0391   0486 

0581 

0676 

0771 

95 

8 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

9 

1813 

1907 

2002 

2096 

2191 

2286 

2380 

2475 

2569 

2663 

, 

PROPORTIONAL  PARTS. 

Diff  .   1 

234 

5 

678 

9 

105   10.5 

21.0    31.5    42.0 

52.5 

63.0    73.5    84.0 

94.5 

104   10.4 

20.8    31.2    41.6 

52.0  !  62.4    72  8    83.2 

93.6 

103   10.3 

20.6    30.9    41.2 

51.5 

61.8    721    82.4 

92.7 

102   10.2 

20.4    30.6    40.8 

51.0 

61.2    714    81.6 

91.8 

101   10.1 

20.2    30.3    40.4 

50.5 

60.6    70  7    80.8 

90.9 

100   10.0 

20.0    30.0    40.0 

50.0 

60.0    70  0    80.0 

90.0 

99    9.9 

19.8    29.7    39.6 

49.5 

59.4    69.3    79.2 

89.1 

i 

34.0 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  460  L.  662.]                                  [No.  499  L.  698. 

T)ifF 

. 

_ 

, 

Ulll. 

460 

662758 

2852 

2947 

3041   31-35 

3230 

3324 

3418 

3512 

3607 

1 

3701 

3795 

3889 

3983  4078 

4172 

4266 

4360 

4454 

4548 

2 

4642 

4736 

4830 

4924  5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

5581 

5675 

5769 

5862  5956 

6050 

6143 

6237 

6331 

6424 

4  i   6518  6612 

6705 

8799 

6892 

6986 

7079 

7173 

7266 

7360 

5    7453  7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

6 

8386  \  8479 

8572 

8665  8759 

8852 

8945 

9038 

9131 

9224 

9317  9410 

9503 

9596  9689 

9782 

9875 

9967 

' 

0060 

0153 

93 

8  670246  0339  0431 

0524  i  0617 

0710 

0802 

0895 

0988 

1080 

9    1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

470 

2098 

2190 

2283 

2375 

2467 

2560 

2652 

2744 

2836 

2929 

1    3021   3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

2    3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

3 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

4 

5778 

5870 

5962 

6053 

6145 

,  6236 

6328 

6419 

6511 

6602 

5 

6694 

6785 

6876 

6968 

7059 

i  7151 

7242 

7333 

7424 

7516 

6    7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

7 

8518 

8609 

8700  8791 

8882 

!  8973 

9064 

9155 

9246 

9337 

91 

8    9428 

9519 

9610 

9700 

9791 

9882 

9973 

OOfi'3 

0154 

024^ 

9  i  680336 

0426 

0517 

0607 

0698 

0789 

0879 

lAJuo 

0970 

1060 

U-*-±O 

1151 

480    1241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

1  i   2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

2    3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

3    3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

4    4845 

4935 

5025 

5114 

5204   5294 

5383 

5473 

5563 

5652 

5    5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

6  !   6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

7  I   7529 

7G18 

7707 

7796 

7'886 

7975 

8064 

8153 

8242 

8331 

cq 

8 

8420 

8oO'J 

8598 

8687 

8776  ;  8865 

8953 

9042 

9131 

9220 

OJ 

9 

9309 

9398 

9486  i  9575 

9664   QTftS 

9841 

9930 

0010 

0107 

490 

690196 

0285 

0373 

0462 

0550 

0639 

0728 

0816 

UUI  J 

0905 

UlUrf 

0993 

1 

1081 

1170 

1258 

1347 

14&5  I 

1524 

1612 

1700 

1789 

1817 

2 

1905 

2053 

2142 

2230 

2318  j 

2406 

2494 

2583 

2671 

2759 

3 

2847 

2935 

3023 

3111 

3199  ! 

3287 

&375 

3463 

8551 

3639 

88 

4 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

5 

4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

6 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

7 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

8 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

9 

8100 

8188 

8275 

8362 

8449  1  8535 

8622 

8709 

8796 

8883 

87 

II 

TROPORTIOXAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7  . 

8 

9 

98 

9.8 

19.6 

29.4 

39.2 

49.0 

58.8 

68.6 

78.4 

88.2 

97 

9.7 

19.4 

29.1 

38.8 

48.5 

58.2 

67.9 

77.6 

87  3 

96 

9.6 

19.2 

28.8 

38.4 

48.0 

57.6  . 

67.2 

76.8 

86.4 

95 

9.5 

19.0 

28.5 

38.0 

47.5 

57.0 

66.5 

76.0 

85  5 

94 

9.4 

18.8 

23.2 

37.6 

47.0 

56.4 

65.8 

75.2 

84.6 

,93 

9.3 

18.6 

27.9 

37.2 

46.5 

55.8 

65.1 

74.4 

83.7 

92 

9.2 

18.4 

27.6 

36.8 

46.0 

55.2 

64.4 

73.6 

82.8 

91 

9.1 

18.2 

27.3 

36.4 

45.5 

54.6 

63.7 

72.8 

81  9 

90 

9.0 

18.0 

27.0 

36.0 

45.0 

54.0 

63.0 

72.0 

81.0 

89 

8.9 

17.8 

26.7 

a<5.6 

44.5 

53.4 

62.3 

71.2 

80  1 

88 

8.8 

17.6 

26.4 

35.2 

44.0 

52.8 

61.6 

70.4 

79  2 

87        8.7 

17.4 

26.1 

34.'8 

43.5 

52.2 

-60.9 

69.0 

7«:3 

86         8.6 

17.2 

25.8 

34.4 

.43.0 

51.6 

60.2 

68.8 

77.4 

TABLE  XXIV.-LOGARITHMS  OF  NUMBERS. 


No.  500  L.  698.1                                   [No.  544  L.  736. 

N. 

0 

1 

2 

8 

4 

5 

6 

7 

8 

9   Diff. 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

-j 

9838 

9924 

0011 

0098 

0184 

0271  0358 

0444 

0531  i  0617 

2 

"TOOToT  0790 

0877 

0963 

1050 

1136   1222 

1309 

1395  1  1482 

3 

1568 

1654 

1741 

1827 

1913 

1999  2086 

2172  2258  i  2344 

4 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

5 

3291 

3377  3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065  |   86 

C 

4151 

4236  i  4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

7 

5008 

5094  i  5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

8 

5864 

5949 

6035 

6120 

6206  i 

6291 

6376 

6462 

6547 

6632 

9 

6718 

6803 

6888 

6974 

7059  i 

7144 

7'229 

7315 

7400 

7485 

510 

7570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

QS; 

1 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015  9100 

9185 

OO 

2 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863  9948 

nnQQ 

3 

710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710  0794  0879 

4 

0963 

1048 

1432 

1217 

1301 

1385 

1470 

1554   1639   1723 

5 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397  2481 

2566 

6 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3323 

3407 

Rl 

7 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

04 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916- 

5000 

508-1 

9 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

520 

6003 

6087 

6170 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

1 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

oo 

3 

8502 

8585 

8668 

8751 

8834  . 

8917 

9000 

9083 

9165 

9248 

oo 

4 

9331 

9414 

9497 

9580  9663  i 

9745 

9828 

9911 

9994 

—  —  — 

5 

720159 

0242 

0325 

0407 

0490  i 

0573 

0655 

0738 

0821 

0903 

6 

0986 

1068 

1151 

1233 

1316  ! 

1398 

1481 

1563 

1646 

1728 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

9 

3456 

8538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

4276 

4358 

4440 

4522 

4604  '• 

4685 

4767 

4849 

4931 

5013 

1 

5095 

5176  5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

2 

5912 

5993  6075 

6156  6238 

6320  |  6401 

6483 

(55(54 

6646 

3 

6727 

6809  !  6890 

6972  7053 

7134  i  7216 

7297 

7379 

7460 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

5 

8:354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

6 

9165 
J9<4 

9246   9327 

9408 

1489 

9570 

9651 

9732 

9813 

9893 

81 

0055  m  cm 

0217  0298 

0378 

0459 

0540 

0621 

0702 

8 

730782 

0863 

0944 

1024   1105  I 

1186 

1266 

1347 

1428 

1508 

9 

1589 

1669 

1750 

1830 

1911  i 

1991 

2072 

2152 

2233 

2313 

540 

2394 

2474 

2555 

2635 

2715  ' 

2796 

2876 

2956 

3037 

3117 

1 

3197 

3278 

3358 

3438 

3518  i 

3598 

3679 

3759 

3839 

3919 

2 

3999 

4079 

4160 

4240  4320  ; 

4400 

4480 

4560 

4640 

4720 

Of) 

3 

4800 

4880 

4960 

5040 

5120 

5279 

5359 

5439 

5519 

OU 

4 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

87    8.7 

17.4    26.1    34.8 

43.5 

52.2    60.9    69.6 

78.3 

86    8.6 

17.2    25.8    34.4 

43.0 

51.6    60.2    68.8 

77.4 

85    8.5 

17.0    25.5    34.0 

42.5 

51.0    59.5    68.0 

76.5 

84    8.4 

16.8    25.2    33.6 

42.0 

50.4    58.8    67.2 

75.6 

347 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  545  L.  736.] 

[No.  584  L.  767. 

1C1 

. 

A 

v   uiu. 

i 

545 

736397  6476  6556 

6635  6715   6795  6874 

6954 

7034  7113 

6 

7193  7272  7352 

743 

1  I  7511  i!  7'590 

7070 

7' 

•40 

7829  1  7908 

7 

7987  8067  j  8146 

8225  i  8305  ||  8384 

8463 

8543 

8022  '  8701 

8 

8781 

8869  8939 

901 

8  9097 

:  9177 

9256 

i 

)35 

9414  |  9493 

9    9572 

9651   9731 

961 

0  9889 

i  GOKS 

0047 

o 

•>r. 

0205  ™>&/i 

7Q 

550  740363 

0442  0521 

0600 

0678-   0757 

0836 

0915 

0994 

1073 

19 

1 

1152 

1230  1  1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

2    1939 

2018  2096 

217 

5  2254 

2332 

2411 

& 

I8!t 

2568 

2647 

3    2725 

2804  2882 

290 

1   3039 

3118 

3196 

S& 

375 

3353 

3431 

4    3510 

3588  ;  3667 

3745  i  3F23 

3902 

3980 

4058 

4136 

4215 

5    4293 

4371 

4449 

452 

8 

4606 

4684 

4762 

4 

vll) 

4919 

4997 

6 

5075 

5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933  i  6011 

608 

0 

6167 

6245 

6323 

* 

101 

6479 

6556 

8 

6034 

0712   6790  68C 

8 

6945 

7'023  7101 

7!) 

7256 

7334 

9 

7412 

7489  7567 

7645 

7722 

7800  |  7878 

7955 

8033 

8110 

560 

8188 

8266  8343 

8421 

8498 

8576  8653 

8731 

8808 

8885 

1 

8963 

9040  9118 

9195 

9272 

9350  9427 

9504 

9582 

9659 

2 

9730 

9814  9891  996 

a 

0045 

'  0123  0200 

0277  !  0354 

0431 

3  i  750508 

0586 

0663 

07'. 

0 

0817 

C894  0971 

1 

)48  1125 

1202 

4  I   1279 

1356 

1433   151 

0 

1587 

1604  1741 

1 

318  1895 

1972 

5 

2048 

2125 

2202 

2279  2356 

2433  2509 

2 

586 

2063 

2740 

77 

6 

2816 

2893 

2970  3& 

7  3123 

3200  |  3277 

ft 

353 

3430 

3506 

7 

3583 

3660 

3736  381 

3 

3889 

3966  i  4042 

4 

119  i  4195 

427'2 

8 

4348 

4425 

4501 

4578 

4654 

4730  i  4807 

4883  1  4960 

50:^6 

9 

5112 

5189 

5265 

5341 

5417 

5494  5570 

5646  5722 

5799 

570 

5875 

5951 

6027 

6103 

6180 

6256 

6332 

6408 

6484 

6560 

1 

6636 

6712 

6788 

ese 

4 

6940 

7016 

7092 

7 

HIM 

7244 

73SO 

76 

2 

7396 

7472 

7548 

7624 

7700 

7775 

7851 

7927 

8003 

8079 

3 

8155 

8230 

8306 

mi 

8 

8458 

8533 

8609 

8 

186 

8761 

8836 

4 

8912 

8988 

9063  9U 

9 

9214 

9290  9366 

!) 

441 

9517 

9592 

5 

9008 

9743 

9819 

'.IS', 

4 

9970 

0045  m  01 

Q 

Kir. 

no  TO 

0347 

6 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0950   1025 

1101 

7 

1176 

1251 

1326 

1402 

1477 

i  1552 

1"627 

1 

102  1778 

1853 

8 

1928 

2003 

2078 

8M 

:>> 

2228 

2303 

2378 

9 

453 

2529 

2604 

9 

2679 

2754 

2829 

2904 

2978 

3053 

3128 

3 

m 

3278 

3353 

75 

580 

3428 

3503  3578 

3653 

3727 

|  3802 

3877 

3 

m 

4027 

4101 

1 

4176 

4251   4326  4400 

4475 

4550  4624 

4699 

4774 

4848 

2 

4923 

4998  i  5072  51^ 

17 

5221 

|  5296  5370 

5 

445 

£520 

5594 

3 

5069 

5743 

5818 

58! 

£ 

5966 

i  6041 

6115 

6190 

6264 

C338 

4 

6413 

6487 

6562 

GO: 

(i 

6710 

;  6785  6859 

1 

6 

m 

7007 

7082 

PROPORTIONAL  PARTS. 

Diff.   1 

2      3 

4 

5 

6 

7      8 

9 

&3    8.3 

16.6    24.9 

33.2 

41.5 

49.8 

58.1    66.4 

74.7 

82    8.2 

16.4    24.6 

32.8 

41.0 

49.2 

57.4    65.6 

73.8 

81    8.1 

16.2    24.3 

32.4 

40.5 

48.6 

56.7    64.8 

72.9 

80    8.0 

16.0    24.0 

32.0 

40.0 

48.0 

56.0    64.0 

72.0 

79    7.9 

15.8    23.7 

31.6 

39.5 

47.4 

55.3    63.2 

71.1 

78    ".8 

15.6    23.4 

31.2 

39.0 

46.8 

54.6    62.4 

70.2 

77    ".1 

15.4    23.1 

30.8 

38.5 

46.2 

53.9    61.6 

69.3 

76    ".6 

15.2    22.8 

30.4 

38.0 

45.6 

53.2    GO.  8 

68.4 

75    7.5 

15.0    22.5 

30.0 

37.5 

45.0 

52.5    60.0 

67.5 

74  |  ".4 

14.8    22.2 

29.6 

37.0 

44.4 

51.8    59.2 

66.6 

348 


TABLE  XXIV. -LOGARITHMS  OF  NUMBERS. 


No.  585  L.  767.] 

[No.  629  L.  799. 

N. 

0 

1 

2 

3 

4 

|  5  i  . 

7 

89   Diff. 

685 

~76715G 

7230  7304 

i  7379 

7453 

7527 

7601 

7675 

7749  7823  i 

6 

7898 

7972  8046 

i  81 

<50  8194 

8268 

8342 

8416 

8490  !  8564    74 

7 

8638 

8712  1  8786 

1  8860  j  8934 

9008 

9082 

9156 

9230  9303 

8 

9377 

9451   9525 

95 

99  9673 

1  974B 

9820 

9894 

0068 

j  Jt/UO  - 

9 

770115 

0189  0263 

0336  0410  j  0484 

0557 

0631 

;  !  0042 
0705  0778 

590 

0852 

0926 

0999 

1073 

1146   1520 

1293 

1367 

1440  !  1514 

1 

1587 

1661   1734 

18 

)8 

1881   1955 

2028 

2102 

2175  2248 

2 

2322 

2395  2468 

25- 

12 

2615   2688 

2762 

2835 

:  2908  2981  i 

3 

3055 

3128  3201 

3274  3348 

i  3421 

3494 

3567 

3640  3713 

4 

3786 

3860  3933 

4006 

4079  !  4152 

4225 

4298 

4371  |  4444    73 

5 

4517 

4590 

4663 

47 

36  i  4809   4882 

4955 

5028 

5100  5173 

6 

5246 

5319 

5392 

5465  j  5538   5610 

5683 

5756 

5829  !  5902 

7 

5974 

6047  6120 

bl 

)3  i  6265   6338 

6411 

6483 

6556  0629 

8 

6701 

6774  6846 

69 

19  6992  !  7'064 

7137 

7209 

7282  i  7354 

9 

7427 

7499 

7572 

7644 

7717 

|  7789 

7862 

793-1 

8006  I  8079 

600 

8151 

8224 

8296 

83 

58  8441 

1  8513 

8585 

8658 

8730  8802 

1 

8874 

8947 

9019 

90 

)1 

9163   9236 

9308  9380 

9452  9524 

2 

9596 

9669 

9741 

U8 

& 

9885 

i  9957 

fiAOO 

3 

780317 

0389  0461 

0533 

0605  ;  0677 

UU/vU 

0749 

0821 

01*3 

0893 

0965 

72 

4 

1037 

1109 

1181 

12, 

S8 

1324 

1396 

1468 

1540 

1612 

1684 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

6 

2473 

2544 

2616 

26* 

>s 

8759 

2831 

2902 

2974 

3046 

3117 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

8 

3904 

3975 

4046 

41 

8 

4189 

4261 

4332 

4403 

4475 

4546 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

1 

6041 

6112  6183 

6£ 

)4  !  6325  t  6396 

6467 

6538 

6609 

6680  !   71 

2 

6751 

6822  6893 

m 

J4 

7035  I  7106 

7177 

7248 

7319  :  7390 

3 

7460 

7531  7602 

7673 

7744  i  7815 

7885 

7956 

8027  80!  )8 

4 

8168 

8239  8310 

Jl  8451   8522 

8593 

8663 

8734 

8804 

5 

8875 

8946  9016 

9087  9157   9228 

9299   9369 

9440 

9510 

6 

9581 

9651   9722 

971 

\2  9863   9933 

AAA  i  1  /vv*  t 

[     t 

7 

790285 

0356  0426 

0496  0567   0637 

UUt>4 

0707 

0778 

0144 

0848 

1»1O 

0918 

8 

0988 

1059   1129 

1199   1269  i  1340 

1410 

1480 

l.")0   1620  ! 

9 

1691 

1761 

1831 

1901 

1971  1  2041 

2111 

2181 

2252 

2322  ; 

620 

2392 

2462  2532 

2602  2672   2742 

2812 

2882 

2952 

3022  :   70 

1 

3092 

3162  3231 

33C 

)1  3371  !  3441 

3511   3581 

3651 

3721  i 

2 

3790 

3860  3930 

4000  i  4070  |]  4139 

4209  4279 

4349 

4418  \ 

3 

4488 

4558  I  4627 

461 

7  4767  \\  4836 

4906  4976 

5045  i  5115 

4 

5185 

5254 

5324 

5393  5463  ||  5532 

5602 

5672 

5741   5811  i 

5 

5880 

5949 

6019 

60£ 

8  6158   6227 

6297 

(5366 

6436   6505 

6 

6574 

6644 

6713 

67> 

2  6852  !i  6921 

6990 

7060 

7129 

7198 

7 

7268 

7337 

7406 

747 

5 

7545  i:  7614 

7683 

7752 

7821 

7890 

8 

7960 

8029 

8098 

8167 

8236   8305 

8374 

8443 

8513 

8582 

9 

8651 

8720 

8789 

885 

6  8927  !  8996  i 

9065 

9134 

9203 

9272 

69 

PROPORTIONAL  PARTS. 

Diff 

1 

2      3 

4 

5 

678 

9 

75 

7.5 

15.0    22 

5 

30.0    37.5 

45.0    52.5    60.0    67.5 

74 

7.4 

14.8    22 

2 

29.6 

37.0 

44.4    51 

.8    59.2 

66.6 

73 

7.3 

14.6    21 

9 

29.2 

36.5 

43.8    51 

.1    58.4 

65.7 

72 

7.2 

14.4    21 

6 

28.8 

36.0 

43.2    50.4    57.6 

64.8 

71 

7.1 

14.2    21 

3 

28.4 

&5.5 

42.6    49 

.7    56.8 

63.9 

70 

7.0 

14.0    21 

0 

28.0 

35.0 

42.0    49 

.0    56.0 

63.0 

69 

6.9 

13.8    20 

7 

27.6 

34.5 

41.4    48 

.3    55.2    62.1 

349 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  630  L.  799.]                                [No.  674  L.  829. 

j 

0 

8     a   T\Z-CP 

. 

J. 

SB 

\ 

\ 

i/m. 

630 

799341 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892  9961 

1 

800029 

0098 

0167 

0236 

0305   0373 

0442 

0511 

0580 

0648  ! 

2  j   0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198  |  1266 

1335 

3  i   1404 

1472 

1541 

1609   1678 

1747 

1815 

1884   1952 

2021  ! 

4  1   2089 

2158 

2226 

2295  i  2363 

2432 

2500 

2568  !  2637  2705 

5    2774 

2842 

2910 

2979  3047 

3116 

3184 

3252  I  3321 

3389 

6  i   3457 

3525 

3594 

3662  !  3730 

3798 

3867 

3935 

4003 

4071 

7    4139 

4208 

4276 

4344 

4412 

4480  4548 

4616* 

4685 

4753 

8    4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433    68 

9    5501 

5569 

5637 

5705 

5773   5841 

5908 

5976  6044 

6112 

640  j  806180 

6248 

6316 

6384 

6451   6519 

6587 

6655  6723 

6790 

1    6858 

6926 

6994 

7061   7129  |i  7197 

7264 

7332 

7400 

7467 

2    7535 

7603 

7670  i  7738  7806  i|  7873 

7941 

8008 

8076 

8143 

3  !   8211 

8279 

8346  I  8414  8481 

8549 

8616 

8684 

8751 

8818 

4 

8886  8953 

9021 

9088 

9156 

9223  9290 

9358 

9425 

9492 

9560  9627 

9694 

(»"*'(!•> 

9829 

9896 

9964 

d 

0031 

0098 

n-jcK 

6 

810938 

0300 

0367 

0434 

0501 

0569 

0636 

0703 

0770 

0837 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

650 

2913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

4 

*5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042  i  6109 

6175 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

6904 

6970 

7036 

7102 

7169 

7235 

7:301 

7367 

7433 

7499 

7 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

D 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660    9544  9610 

9676  9741 

9807 

9873 

9939 

fl070 

niqc 

1 

820201 

0267 

0333  I  0399 

0464 

0530 

0595 

0661 

UUiU 

0727 

UloO 

0792 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

7 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4(546 

4711 

£>K 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

DO 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464* 

6528 

6593 

6658 

1 

6723 

6787 

6852 

6917  6981 

7046 

7111 

7175 

7240 

7305 

2 

7369 

7434 

7499 

7563  7628 

7692 

7757 

7821 

7886 

7951 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

PROPORTIONAL  PARTS. 

Diff.   1 

2-34 

5 

678 

9 

68    6.8 

13.6  1  20.4    27.2 

34.0 

40.8    47.6    54.4 

61.2 

67    6.7 

13.4  1  20.1    26.8 

33.5 

40.2    46.9    53.6 

60.3 

66    6.6 

13.2    19.8    26.4 

33.0 

39.6    46.2    52.8 

59.4 

65    6.5 

13.0    19.5    26.0 

32.5 

39.0    45.5    52.0 

58.5 

64    6.4 

IS.  8    19.2    25.6 

32.0 

38.4    44.8    51.2 

57.6 

350 


TABLE  XXIV.— LOGARITHMS   OF  NUMBERS. 


No.  675  L.  829.] 

[No.  719  L.  857. 

N. 

0 

1 

2 

8 

9   !  Diff. 

I 

675 

829304 

9368  9432 

9497  9561  !!  9625 

9690 

9754 

9818 

9882 

6 

9947 

•  

0011  i  0075  Olu 

*o  i  ncvM  I  noco   nQQo  i  rwo« 

7 

830589 

0053  0717  0781   0845  I  0909  0973  i  1037 

1102 

1106 

8 

1230 

1294   1358  !  14S 

J2  1480   1550 

1014   1078 

1742 

IfcOG  i   64 

9 

1870 

1934   1998 

99 

>2  2120  '  2189 

2253 

2317 

2381 

2445 

680 

2509 

2573  2637 

2700  2764   2828  2892  2956 

3020 

S083  i 

1 

3147 

3211 

3275 

3338  3402  ||  3400  3530  3593 

3657 

3721  i 

2 

3784 

3848 

3912 

39" 

"5  4039 

4103  41C6 

4230 

4294 

4357  i 

3 

4421 

4484 

4548 

4011   4075 

I  4739   4802 

4HGO 

4929 

4993  j 

4 

5056 

5120 

5183 

FrtjX, 

17   5310 

5373  5437  !  5500 

5504 

5027 

5 

5691 

5754 

5817 

5881   5944 

6007  0071   0134 

0197 

0201 

6 

6324 

6387 

6451 

65] 

4 

6577 

6641   6704  !  6767 

0830 

0894 

7 

6957 

7020 

7'083 

71^ 

10 

7210 

i  7273 

7336  |  7399 

7402 

7525 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7'967  8030 

8093 

8156 

9 

8219 

8282 

8345  8408 

8471 

8534  8597 

8000 

8723 

8786    63 

690 

8849 

8912 

8975   90; 

B 

9101 

9164  9227  9289 

9352 

9415 

1 

9478 

9541 

9004  9007 

9729  j  9792   9855   9918 

9981 



0043 

2 
3 

840106 
0733 

0169 
0790 

0232 
0859 

0294 
0921 

0357   0420 
0984   1046 

0482  0545 
1109  1172 

0008 
1234 

0671 
1297 

4 

1359 

1422 

1485 

1547 

1010   1672 

1735 

1797 

1800 

1922 

5 

1985 

2047 

2110 

21' 

a 

2235 

2297 

2200 

2422 

2484 

2547 

6 

2609 

2072 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

8170 

7 

3233 

3295 

3357 

34i 

.'(.) 

3482 

3544 

3006 

3009 

3731 

3793 

8 

3855 

3918 

3980 

4042 

4104 

4100 

4229 

4291 

4353 

4415* 

9 

4477 

4539 

4001 

4604 

4726 

4788 

4850 

4912 

4974 

5036 

700 

5098 

51GO 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

62 

1 

5718 

5780 

5842 

59( 

14 

5966   6028 

0090 

0151 

0213 

0275 

2 

6337 

0399 

0401 

6523 

6585   6046 

6708 

0770 

0832 

6894 

3 

6955 

7017 

7079 

71, 

1 

7202  '  7264 

7326 

7388 

7449 

7511 

4 

7573 

7634 

7096 

7758 

-7819   7881 

7943 

8004 

80GG 

8128 

5 

8189 

8251 

8312 

83" 

4 

8435  , 

8497 

8559 

8020 

8082 

8743 

6 

8805 

8800 

8928 

Kfr 

B 

9051 

9112 

9174 

9235 

9297 

9358 

r 

9419 

9481 

9542   9004 

9665 

9726 

9788 

9849 

9911 

9972 

8 

850033 

0095 

0156  0217 

0279  i 

0340 

0401 

0402 

0524 

0585 

9 

0046 

0707 

0709  0830 

0891  | 

0952 

1014 

1075 

1130 

1197 

710 

1258 

132.0 

1381   1442 

1503  j 

1564 

1625 

1686 

1747 

1809 

1 

1870 

1931 

1992  SOt 

a 

2114 

2175 

2236 

2297 

2358 

2419 

2 

2480 

2541 

2602 

2«( 

.3 

27'24  i 

2785 

2846 

2907 

25)08 

3029 

61 

3 

3090 

3150 

3211  £27 

2  3333 

3394 

3455 

3516 

3577 

2037 

4 

3698 

3759 

3820  38£ 

1 

3941 

4002 

4063 

4124 

4185 

4245 

5 

4306 

4307 

44-28  4488  4549  i 

4610 

4670 

4731 

4792 

4858 

6 

4913 

4974 

5034  50S 

5  5150 

5216 

5277 

5337 

5398 

5459 

7 

5519 

5580 

5640  5701  5761 

5822 

5882 

5943 

0003 

6064 

8 

6124 

6185 

6245 

63C 

G 

6366  : 

6427 

6487 

6548 

OG08 

CG68 

9 

6729 

6789 

6850 

6910 

6970  i 

7031 

7091 

7152 

7212 

7272 

PROPORTIONAL  PARTS. 

Diff.   1 

2      3 

4 

5 

6  . 

r 

8 

9 

65    6.5 

13.0    19.5 

2G.O 

32.5 

.39.0    45.5 

52.0 

58.5 

64    6.4 

12.8    19.2 

25.  G 

32.0 

38.4    44 

.8 

51.2 

57.6 

63    6.3 

12.6    18.9 

25.2 

31.5  i  37.8    44 

.1 

50.4 

56.7 

62    6.2 

12.4    18.6 

24.8 

31.0    37.2    43.4  . 

49.6 

55.8 

61    6.1 

12.2    18.3 

24.4 

30.5 

36.6    42 

.7 

48.8 

54.9 

60    6.0 

12.0    18.0 

24.0 

30.0 

36.0    42.0 

48.0 

54.0 

351 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  720  L.  857.]                                  [No.  764  L.  883. 

N. 

0 

1 

2 

3 

4 

6 

6    7 

8 

9 

Diff. 

720 

857332 

7393  7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

1 

7935 

7995 

8056 

8116 

8170 

8236 

8297 

8357 

8417 

8477 

0 

8537 

8597 

8s557 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

3 

9138 

9198  I  9258 

9318 

9379 

\  9439 

9499 

9559 

9619 

9679 

60 

A 

9739 

97'99  9859 

9918 

9978 

0038 

0098 

01^8 

0218 

0278 

5 

860338 

0398  0458 

0518 

057d 

0637 

0697 

0757 

0817 

0877 

6 

0937 

0996 

1056 

1116 

1170 

123*5 

1295 

1355 

1415 

1475 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3114 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

4 

5690 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

5    6287 

6346 

6405 

6465 

6524 

6583 

6642 

67'01 

6760 

6819 

6    6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7    7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

8  ;   8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

9  i   8044 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

91,3 

740    9232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

1 

9818 

9877 

9935 

9994 

0053 

0111 

0170 

ftoos   noc"? 

0345 

2 

870404 

0462 

0521 

0579 

06:38 

0696 

0755 

0813 

0872 

0930 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

4  i   1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

5    2150 

2215 

2273 

2331 

2389 

2448 

250(5 

2564 

2622 

2681 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

7    3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

8  1   3902 

3969 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

442-4 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

750 

5061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

1 

5640  5698 

57'56 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

2 

6218 

0270 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

5 

7947 

8004 

8062 

8119 

8177 

82:34 

8292 

8349 

8407 

8464 

6 

8522 

8579 

8037 

8094 

8752 

8809 

8866 

8924 

8981 

9039 

7 

9098 

9153 

9211 

9208 

9325 

9383 

9440 

9497 

9555 

9612 

g 

9669 

9726 

9784 

9841 

9898 

9956 

0013 

0070 

,  .  „ 

9 

8802-42 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

760 

0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

4 

3093 

3150 

32C7 

3264 

3321 

3377 

34:34 

3491 

3548 

3605 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3      4 

5 

678 

9 

59 

5.9 

11.8 

17.7    23.6 

29.5 

35.4    41.3    47.2 

53.1 

58 

5.8 

11.6 

17.4    23.2 

29.0 

34.8    40.6    46.4 

52.2 

57 

5.7 

11.4 

17.1    22.8 

28.5 

34.2    39.9    45.6 

51  3 

56 

5.6 

11.2 

16.8    22.4 

28.0 

33.6    39.2    44.8 

50.4 

353 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  765  L.  883.]                                 [No.  809  L.  908. 

N. 

0 

1 

2 

Lfi 

1 

!    _ 

!  6 

6 

7 

8 

9 

Diff. 

765 

883661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

7 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

8 

5361 

5418 

5474 

5531 

5587  1  5644 

5700 

5757 

5813 

5870 

9 

5926 

5983 

6039 

6096 

6152   6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

2 

7617 

7674' 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

5 

9302 

9358 

9414 

9470 

9526 

9582 

%38 

9694 

9750 

9806  j   56 

6 

9862 

9918 

9974 

0030 

C086 

i  _1  ... 

01  Q7 

AOAO 

f.npr  I 

7 

890421 

0477 

0533 

0589 

0645 

0700 

uiy  < 
0756 

0812 

\JO\J\J 

0868 

0924 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

1 

2651 

2707 

27'62 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

47'59 

4814 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6907 

7022 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

790 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

55 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

3 

4 

9273 
9821 

9328 
9875 

9383 
9930 

9437 
9985 

9492 

9547 

9602 

9656 

9711 

9766 

5 

0586 

0640 

0695 

0749  0804 

0859 

900367 

0422 

0476 

0531 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

9 

2547 

2601 

2655 

2710 

2764 

2818 

287'3 

2927 

2981 

3036 

800 

3090 

3144 

3199 

3253 

3307 

asei 

3416 

3470 

3524 

3578 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

. 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7'304 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

9 

7949 

8002 

8056 

8110 

8163  II  8217 

8270 

832-1 

8378 

8431 

II          ! 

PROPORTIONAL,  PARTS. 

Diff.   1 

234 

5 

678 

9 

57    5.7 

11.4    17.1    22.8 

28.5 

34.2    39.9    45.6 

51.3 

56    5.6 

11.2    16.8    22.4 

28.0 

33.6    39.2    44.8 

50.4 

55    5.5 

11.0    16.5    22.0 

27.5 

33.0    38.5    44.0 

49.5 

54    5.4 

10.8    16.2    21.6 

27.0 

32.4    37.8    43.2 

48.6 

353 


TABLE  XXIV.-LOGARITHMS  OF  NUMBERS. 


No.  810  L.  908.] 

[No.  854  L.  931. 

N. 

0 

1 

2 

I 

4 

5    6 

7 

8 

9 

Diff. 

810  908485 

8539 

8592 

8646 

8699   8753 

8807  8860 

8914 

8967 

1    9021 

9074 

9128 

918 

1 

9235  i 

9289 

9342  9S 

96 

9449   9503 

2    9556 

9610 

9663  971 

J 

9770  i  9823 

9877  9£ 

•'iit 

9081 

nrwr 

3  910091 

0144 

0197 

0251   0304   0358  0411   0464 

0518 

0571 

4    0624 

0678 

0731 

0784 

0838   0891 

0944 

(!< 

'.IS 

1051 

1104 

5 

1158 

1211 

1264 

131 

7 

1371 

1424 

1477 

i; 

30 

1584 

1637 

6 

1690 

1743 

1797 

185 

) 

1903 

1956 

2009 

i 

c,: 

2116 

2169 

2222 

2275 

2328 

2381 

24:35 

2488 

2541 

2594 

2647 

2700 

8 

2753 

2806 

2859 

291 

3 

2966 

8D19 

3072 

31 

25 

3178 

3231 

9 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

53 

820 

3814 

3867 

3920 

397 

•3 

4026 

4079 

4132 

4184 

4237 

4290 

1 

4343 

4396 

4449 

4502 

4555 

4608 

4C60 

4713 

4766 

4819 

2    4872 

4925  4977 

503 

q 

5083  ;  5136  5189 

• 

341  i  5294 

5347 

3  !   5400 

5453  5505 

555 

s 

5611  1!  5664 

5716 

,-, 

"69  5822 

5875 

4 

5927 

5980   6033 

60&5 

6138  j!  6191 

6243 

6296  6349 

6401 

5 

6454 

6507 

6559 

661 

2 

6664  !  6717 

6770 

( 

<22  6875 

6927 

6 

6980 

7033 

7085 

713 

8 

7190   7243 

7295 

7 

348  7400 

7453 

7 

7506 

7558 

7611 

7663 

7716   7768 

7820 

7873  7925 

7978 

8 

8030 

8083 

81  a5 

818 

8 

8240   8293 

8345 

8 

397  8450 

8502 

9 

8555 

8607 

8659 

8712 

8764   8816 

8869 

8921 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287   9340 

9392 

9444 

94G6 

9549 

9601 

9653 

9706  °r-'K 

j^ 

9810   S862 

9914 

g 

367 

1  ^ 

I  nmo 

0071 

2 

920123 

0178 

0228 

0280  0332  1  0384 

0436 

0489  0541 

UUrf  1 

0593 

3 

0645 

0697 

0749 

08C 

1 

0853 

0906 

0958 

1 

HO 

1062 

1114 

xo 

4 

1166 

1218 

1270 

132 

2 

1374 

1426 

1478 

1 

•530 

1582 

1634 

066 

5 

1686 

1738 

1790 

1842 

1894  j  1946 

1998 

2050 

2102 

2154 

6 

2206 

2258 

2310 

23G 

2 

2414  |l  24C6 

2518 

2 

~'™ 

2622 

2674 

7 

2725 

2777 

2829 

28F 

1 

2933  !  2985 

3037 

S 

3140 

3192 

8 

3244 

3296 

3348  3399 

3451  II  3503 

3555 

8607 

3C58 

3710 

9 

S7'62 

3814 

3865  !  3917 

3969 

4021 

4072 

4124 

4176 

4228 

840 

4279 

4331  4383  4434 

4486 

4538 

4589 

4641 

4693 

4744 

1 

4796 

4848  i  4899 

4951 

5003 

5054 

5106 

5157 

£209 

5261 

2 

5312 

5364 

5415 

54£ 

5518 

i  5570 

5621 

£ 

m 

57'25 

5776 

3 

5828 

5879 

5931 

59F 

1 

6034 

C085 

6137 

(i 

188 

6240 

6291 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

(i 

702  6754 

ceos 

5 

6857 

6908 

6959 

701 

1 

7062 

7114  7165 

7 

<>16 

7268 

7319 

6 

7370 

7422 

7473 

75$ 

4 

7576 

7627 

7'678  7 

7KO 

7781 

7'822 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191  8242 

W93 

8345 

8 

8396 

8447 

8498 

854 

9 

8601 

8652 

8703 

8 

754  8805 

8857 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266  j  9317 

93C8 

850 

9419 

9470 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

9930 

9981 

51 

fW}O 

/V)L 

0 

„ 

n-ioe 

P9<*fi 

o 

tea 

0338 

0389 

2 

930440 

0491 

0542 

UUr 

osr 

i 

0643 

UloO 

0694 

0745 

0796 

0847 

0898 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

J865 

1915 

PROPORTIONAL  PARTS. 

Diff.   1 

2      3 

4 

5 

6 

7 

8 

9 

53    5.3 

10.6    15.9 

21.2 

26.5 

31.8 

37.1 

42.4 

47.7 

52    5.2 

10.4    15.6 

20.8 

26.0 

31.2 

36.4 

41.6 

46.8 

51    5.1 

10.2    15.3 

20.4 

25.5 

30.6 

35.7 

40.8 

45.9 

50    5.0 

10.0    15.0 

20.0 

25.0 

30.0 

35.0 

40.0 

45.0 

TABLE  XXIV. -LOGARITHMS  OF  NUMBERS 


No.  855  L.  931.1 

[No.  899  L.  954. 

1 

1 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

855  931966 

2017  2068 

2118  2169 

2220  2271 

2322  2372 

2423 

6    2474 

2524  2575 

26 

26   2677   2727  2778 

2829 

28' 

0 

2930 

7    2981 

3031   3082 

3133  3183  !  3234  3285 

3335  13386   3437 

8 

3487  3538  3589 

3C 

39  3090   3740  3791 

3841  i  38! 

)2  3943 

9 

3993  4044  4094 

4145  4195   4246  4296 

4347  4397  4448 

860 

4498  4549  4599 

4650  4700   4751  4801 

4852  4902  4953 

1 

5003  5054  !  5104 

5154 

5205  ||  5255  5306 

5356 

5406  :  5457 

2 

5507  5558  5608 

n 

58 

5709 

5759  5809 

5860 

59 

0 

5960 

3 

6011 

6061 

6111 

61 

(52 

6212 

6262  6313 

6363 

64 

3 

6463 

4 

6514 

6564 

6614 

6665 

6715 

!  6765  6815  '• 

6865 

69] 

(.; 

6966 

5 

7016 

7066 

7116 

7167 

7217 

I  7267  7317 

7'367 

7418 

7468 

6 

7518 

7568 

7618 

76 

68 

7718 

i  7769  7819 

7869  79] 

e 

7969 

7 

8019 

8069 

8119 

81 

69 

8219 

8269 

8390 

8370  84i 

•0 

8470 

50 

8 

8520 

8570 

8620 

8670 

8720 

8770 

8820  i 

8870  8920 

8970 

9 

9020 

9070 

9120 

91 

70 

9220 

9270 

9320 

9369  9419 

9469 

870 

9519 

9569  9619 

9669  9719 

9769 

9819 

9S69  9918 

9968 

1 

940018 

0068  1  0118 

0168  |  0218 

;  0267  0317 

0367 

0417 

0467 

2 

0516 

0566  i  0616 

06 

66  !  0716 

0765 

0815 

0865 

09] 

5 

0964 

3 

1014 

1064 

1114 

1163  !  1213 

1263 

1313  ; 

1362" 

1412 

1462 

4 

1511 

1561 

1611 

16 

60  1710 

1760 

1809  : 

1859 

19( 

)<) 

1958 

5 

2008 

2058 

2107 

2157  2207 

2256 

2306 

2355 

2405 

2455 

6 

2504 

2&54 

2603 

26 

58 

2702 

2752 

2801 

2851 

29( 

H 

2950 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

8 

3495 

3544 

3593 

36 

43 

3692 

3742 

3791 

3841 

88! 

0 

3939 

9 

3989 

4038 

4088 

4137 

4186 

|  4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

48.28 

4877 

4927 

4976 

5025 

5074 

51 

24 

5173   5222 

5272 

5321 

537 

0 

5419 

2 

5469 

5518 

5567 

5616 

5665  ||  5715 

5764  | 

5813  5862 

5912 

3 

5961 

6010 

6059 

61 

)8 

6157 

6207 

6256 

6305 

63c 

4 

6403 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747  i 

6796 

6845 

6894 

5 

6943 

6992  i  7041 

70 

X) 

7139 

7189 

7238 

7287 

733 

8 

7385 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7826 

7875 

49 

7 

7924 

7973 

8022 

80 

70 

8119 

8168 

8217 

8266 

831 

5 

8364 

8    8413 

8462 

8511 

8560 

8608 

8657 

8706  i 

8755 

8804 

8853 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292  9341 

890 

1" 

9390 

QS7R 

9439 
9926 

9488 
9975 

9536 

9585 

9634 

9683 

9731  j  9780  9829 

y~<  o 

0024 

0073  |l  0121 

0170  i 

0219  0267 

0316 

2 

950365 

0414 

0462 

0511 

0560  !  0608 

0657 

0706  0754 

0803 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

4 

1338 

1386 

1435 

141 

•tf 

1532 

1580 

1629 

1677 

172 

6 

1775 

5 

1823 

1872 

1920 

1969 

2017  !  2066 

2114 

2163  2211 

2260 

6 

2308 

2356 

2405 

24 

$8 

2502  !  2550 

2599 

2647  269 

6 

2744 

7 

2792 

2841  2889 

29 

W 

2986  i  3034 

3083 

3131   318 

0 

322H 

8 

3276 

3325 

3373 

3421 

3470  i  3518 

3566 

3615  366 

3 

3711 

9 

3760 

3808 

3856 

9 

)5 

3953   4001 

4049 

4098 

414 

S 

4194 

1 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3 

4 

5 

6 

7 

8 

9 

51 

5.1 

10.2 

15.3 

20.4 

25.5 

30.6 

35  7 

40.8    45  Q 

50 

5.0 

10.0 

15.0 

20.0 

25.0 

30.0 

&5.0 

40  0 

450 

49 

4.9 

9.8 

14.7 

19.6 

24.5 

29.4 

34.3 

39.2 

44.1 

48    4.8    9.6    14.4 

19.2 

24.0 

28.8 

33.6 

38.4 

43.2 

355 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No  900  L.  954.1 

[No.  944  L.  975. 

I 

i 

7 

8 

N.    0 

1    2 

I 

t 

4 

5 

6 

9 

Diff. 

900 

954243 

4291  .4339 

4387 

4435 

4484 

4532  4580  4628 

4677 

1 

4725 

4773 

4821 

4869 

4918 

4966 

5014  5062  5110  f  5158 

2 

5207 

5255 

5303 

53, 

51 

5399 

5447 

5495  5 

543  559 

2  5640 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976  6024  6072  \  6120 

4 

6168 

6216 

6265 

63 

8 

6361 

6409 

6457  6 

505  655 

3  6601 

5 

6649 

6697 

6745 

6793 

6840  l  6888 

6936 

6984  7032 

7080 

48 

6 

7128 

7176 

7224 

72 

8 

7320 

i  7368 

7416 

7 

164 

751 

z 

7559 

7 

7607 

7655 

7703 

77 

H 

7799 

7847 

7894 

7 

M2 

799 

) 

8038 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

9    8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

1 
2 

9518 
9995 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

0042  0090 

0138 

0185 

0233 

0280  0 

L'S 

0376  0423 

3 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

4 

0946 

0994 

1041 

10 

u 

1136 

1184 

1231 

1 

.279 

132 

B 

1374 

5 

1421 

1469 

1516 

15 

53 

1611 

1658 

1706 

1 

1W» 

180 

l 

1848 

6 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

7 

2369 

2417 

2464 

25 

11 

2559 

2606 

2653 

2 

roi 

274 

s 

2795 

8 

2843 

2890 

2937 

29 

<5 

3032 

3079 

3126 

3 

174 

322 

1 

3268 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

369 

3 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

1 

4260 

4307 

4354 

44 

)1 

4448 

4495 

4542 

4 

590 

463 

7 

4684 

2 

4731 

4778 

4825 

48 

a 

4919 

4966 

5013 

5 

061  :  510 

8 

5155 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

4 

5672 

5719 

5766 

58 

3 

5860 

5907 

5954 

& 

101 

604 

B 

6095 

47 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

6 

6611 

6658 

6705 

67 

>2 

6799 

6845 

6892 

6 

l.-,9 

698 

; 

7033 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

8 

7548 

7595 

7642 

76 

« 

7735 

7782 

7829 

7 

^75 

792 

2 

7969 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

839 

[) 

8436 

930 

8483 

8530 

8576 

86, 

• 

8670 

8716 

8763 

8810 

8856 

8903 

1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

2 

9416 

9463 

9509 

95, 

>(j 

9602 

9649 

9695 

9 

978 

1 

9835 

3 

9882 

9928 

OQ7* 

OOL 

>1   nru:e 

0114 

0161 

0* 

)H7 

025<x 

< 

0300 

4 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

5 

0812 

0858 

0904 

09f 

>1 

0997 

1044 

1090 

1 

137 

118, 

5 

1229 

6 

1276 

1322 

1369 

1415 

1461 

1508  1554   1601 

1647 

1693 

7 

1740 

1786 

1832 

18r 

'9 

1925 

1971 

2018 

2( 

)<>4 

211 

) 

2157 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

asis 

8869 

3405 

3451 

3497 

3543 

1 

3590 

3636 

3682 

37$ 

« 

3774  1 

3820 

3866 

» 

)13 

395< 

) 

4005 

2 

4051 

4097 

4143 

4*1 

1 

4235  ! 

4281 

4327 

4. 

04 

442( 

) 

4466 

3 

4512 

4558 

4604 

4& 

0 

4696 

4742 

4788 

488( 

) 

4926 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

PROPORTIONAL  PARTS. 

Diflf.   1 

2      3 

4 

5 

6 

7 

8 

9 

47    4.7 

9.4    14.1 

18.8 

23.5 

28.2 

32.9 

37.6 

42.3 

46   .4.6 

9.2    13.8 

18.4 

23.0 

27.6 

32.2 

36.8 

41.4 

35G 


TABLE  XXIV.— LOGARITHMS  OF  NUMBERS. 


No.  945  L.  975.]                                  [Xo.  989  L.  995. 

N. 

0 

1 

2 

3 

4 

5 

C 

7 

6 

9 

Diff. 

945 

975432  5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

6 

5891'  £.937 

5983 

6029 

6075 

6121 

6167 

6212 

6258  ;  6304 

7 

6350  6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717  6763 

8 

6808  j  6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7'220 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

1 

8181  i  8226  8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

2 

8637  8683  j  8728 

8774 

8819 

8865 

8911 

8956 

8002 

9047 

3 

9003  !  9138  9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

4 

95-18  9594  9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

5 

980003  C049  I  0094 

0140 

0185 

0231 

0276 

0322 

0367  0412 

G 

(458  0503  ;  0549 

0594 

0640 

0685 

0730 

0776 

Ob21  !  0867 

7 

0912  :  0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275   1320 

8 

1366  ;  1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

1 

2723  ;  2769 

2814 

2859 

2904 

2949 

2994 

3040 

8085 

3130 

2 

3175  3220 

3265 

3310 

3356 

3401 

3446 

3491- 

3536 

3581 

3 

3626  3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

4 

4077  !  4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

5 

4527  4572 

4617  4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977  5022 

5067  5112 

5157 

5202 

5247 

5292 

5337 

5382 

7 

5426  5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

8 

5875  5920 

5965 

C010 

C055 

6100 

6144 

6189 

6234 

6279 

9 

6324 

6369 

6413 

6458 

6503   6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

1 

7219 

7264 

7--:09 

7353 

7398 

7443 

7488 

7'532 

7577 

7622 

2 

7666 

7711 

7756 

7800 

7846 

7890 

7934 

7979 

8024 

8068 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

6 

0450 

9895 

9494 
9939 

9539 

onOQ 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

t7jOO 

C028 

0072 

0117 

0161 

0206 

0250 

0294 

8 

990389 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

1 

16G9 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

5 

3436 

3480 

&524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

G 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

9 

5196  5240 

5284 

5328 

5372 

5416 

5460 

55U4 

5547 

5591 

PROPORTIONAL  PARTS. 

Diff 

1 

234 

5 

G      7      8 

9 

46 

4.6 

9.2    13.8    18.4 

23.0 

27.6    32.2    36.8 

41.4 

45 

4.5 

9.0    13.5    18.0 

22.5 

27.0    31.5    36.0 

40.5 

44 

4.4 

8.8    13.2    17.6 

22.0 

26.4    30.8    35.2 

39.6 

43 

4.3 

8.6    12.9    17.2 

21.5 

25.8    30.1    34.4 

38.7 

357 


TABLE  XXIV.- LOGARITHMS  OF  NUMBERS. 


No.  990  L.  995.] 

[No.  999  L.  999. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

990     995G35 

5679 

5723 

5767     5811 

5854 

5398 

5942 

5986     6030 

1  I       6074 

6117 

6161 

6205  I  6249 

6 

293 

6:337 

b3b 

.) 

6424     6468 

44 

2 

6512 

6555 

6599 

6643 

6687 

(i 

731 

6774 

681 

8 

6862 

6906 

3 

6949 

6993 

7037 

7080 

7124 

3 

168 

7212 

7255 

7299 

7343 

4 

7386 

7430 

7474 

7517 

7561 

3 

605 

7648 

76(J 

2 

7736 

7779 

7823 

7867 

7910 

7954 

7998 

8 

041 

8085 

812 

'.» 

8172 

8216 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

rr 

8695 

8739 

8782 

8826 

8869 

8 

913 

8956 

90C 

6 

904S 

9087 

8 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

947£ 

9522 

9 

9565 

9609 

9C52 

9696 

9739 

9783 

9826 

9870 

991c 

t     9957 

43 

LOGARITHMS  OF  NUMBERS 

FROM  1  TO 

100. 

N. 

Log. 

1 
N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0.000000 

21 

1.522219 

41 

1 

.612784 

61 

1.785330 

81 

1.908485 

2 

0.301 

m 

22 

1.342423 

42 

1 

.6 

282  1 

) 

62 

j_ 

•'.):. 

392 

82 

1.913814 

3 

0.477121 

23 

1.361728 

43 

1 

.633468 

63 

1^799341 

83 

1.919078 

4 

0.602 

)60 

24 

1.380211 

44 

1 

.6 

1845 

J 

64 

1. 

<i)t 

180 

84 

1.924279 

5 

0.698 

}70 

25 

1.397940 

45 

1 

.653213 

65 

1.812913 

85 

1.929419 

6 

0.778151 

26 

1.414973  ! 

46 

1 

.662758 

66 

1.819544 

86 

1.934498 

7 

0.845( 

HIS 

27 

1.431364  I 

47 

1 

.6 

raog 

i 

67 

1. 

•yl 

.075 

87 

1.939519 

8 

0.903090 

28 

1.447158 

48 

1 

.681241 

68 

1.832509 

88 

1.944483 

9 

0.954 

.'43 

29 

1.462398 

49 

1 

.6 

(01!) 

\ 

69 

1. 

<fr 

849 

89 

1.949390 

10 

1.000000 

30 

1.477121 

50 

1 

.698970 

70 

1.845098 

9J 

1.954243 

11 

1.041393 

31 

1.491362 

51 

1 

.707570 

71' 

1.851258 

91 

1.959041 

12 

1.079 

181 

32 

1.505150  ! 

52 

1 

.7 

KM  HI 

} 

72 

1  _ 

332 

92 

1.963788 

13 

1.113 

)43 

33 

1.518514 

53 

1 

j 

2427 

3 

73 

i! 

s 

323 

93 

1.968483 

14 

1.146128 

34 

1.531479 

54 

1 

'7 

3389 

i 

74 

1.869232 

94 

1.973128 

15 

1.176091 

35 

1.544008 

55 

1 

3 

4036, 

} 

75 

1.875061 

95 

1.977724 

16 

1.204120 

36 

1.556302 

56 

1 

7 

48188 

76 

1.880314 

96 

1.982271 

17 

1.230 

149 

37 

1.568202 

57 

1 

'.7 

5587 

> 

77 

1. 

•<s« 

191 

97 

1.986772 

18 

1.255273 

38 

1.579784  ! 

58 

1 

.763428 

78 

1.8C2195 

98 

1.991226 

19 

1.278 

~54 

39 

1.591065  i 

59 

1 

.7 

70S5 

I 

79 

1_ 

s'*i  i 

(527 

99 

1.995635 

20 

1.301030 

40 

1.602060 

60 

1.778151 

80     1. 

)<>: 

090 

100 

2.000000 

••;. 

Value 

at  0°. 

Sign 
in  1st 
Quad. 

ValiK 
at  90° 

>     Sign 
5    in2d 
•    Quad. 

Value 
at 
180°. 

Sign 
in3d 
Quad. 

Value 
at 
270°  - 

Sign 
in  4th 
Quad. 

Value 
nt 

360°. 

Sin 

_j_ 

R          a. 

o 

R 

o 

Tan  
Sec  

8 

f 

00 
CO 

1 

0 

R 

+ 

co 

CO 

4- 

0 
R 

Versin.... 
Cos  

0 
R 

R 
0 

+ 

2R 
R 

+ 

R 
0 

1 

O 

R 

Cot  

GO 

_i_ 

0 



co 

i 

O 

CO 

Cosec  

CO 

+ 

R 

+ 

CO 

—  ' 

R 

— 

00 

R  signifies  equal  to  rad  ;  co  signifies  infinite  ;  O  signifies  evanescent. 

353 


TABLE  XXV. -LOGARITHMIC  SINES 


179' 


" 

> 

Sine. 

q-l 

Tang. 

Cotang. 

q  +  l 

Dl" 

Cosine. 

/ 

4.685 

15.314 

0 

0 

Inf.  neg. 

575 

|575 

Inf.  neg. 

Inf.  pos. 

425 

ten 

60 

60 

1 

6.463726 

575 

575  6.463726 

13.536274 

425 

ten 

59 

120 

2 

.764756 

575 

08 

.764756 

.235244 

425 

ten 

58 

180 

3 

6.940847 

575 

575 

6.940847 

13.059153 

425 

ten 

57 

240 

4 

7.065786 

575 

575 

7.065786 

12.934214 

425 

ten 

56 

800 

5 

.162696 

575 

575 

.162696 

.837304 

425 

ten 

55 

360 

6 

.241877 

575 

575 

.241878 

.758122 

425 

.02 

9.999999 

54 

420 

7 

.308824 

575 

575 

.308825 

.691175 

425 

.00 

.999999 

53 

480 

.366816 

574  I  576 

.366817 

.633183 

424 

.00 

.999999 

52 

540 

9 

.417968 

574 

576 

.417970 

.582030 

424 

.00 

.999999 

51 

GOO  10 

.463726 

574 

576 

.463727 

.536273 

424 

.02 

.999998 

50 

660  11 

7.505118 

574 

^576 

7.505120 

12.494880 

434 

.00 

9.999998 

49 

720   12 

.542906 

574 

577 

.542C09 

.457091 

423 

.02 

.999997 

48 

780  13 

.577668 

574 

•  577 

.577672 

.422328 

423 

.00 

.999997 

47 

840 

14 

.609853 

574 

577 

.609857 

.390143 

423 

.02 

.999996 

46 

900 

15 

.639816 

573 

:578 

.639820 

.360180 

422 

.00 

.9999% 

45 

960 

16 

.667845 

573 

578 

.667849 

.332151 

422 

.02 

.999995 

44 

1020 

17 

.694173 

573 

578 

.694179 

.305821 

422 

.00 

.999995 

43 

1080 

18 

.718997 

573 

;  579 

.719003 

.280997 

421 

.02 

.999994 

42 

1140 

19 

.742478 

573 

579 

.742484 

.257516 

421 

.02 

.999993 

41 

1200 

20 

.764754 

572 

580 

.764761 

.235239 

420 

.00 

.999993 

40 

1260 

21 

7.785943 

572 

J580 

7.785951 

12.214049 

420 

.02 

9.999992 

39 

1320 

22 

.806146 

572 

:  581 

.806155 

.193845 

419 

.02 

.999991 

38 

1380 

23 

.825451 

572  j  581 

.825460 

.174540 

419 

.02 

.999990 

37 

1440 

24 

.843934 

571 

582 

.843944 

.156056 

418 

.02 

.999989 

36 

1500 

25 

.861662 

571 

583 

.861674 

.138326 

417 

.00 

.999989 

35 

1560 

26 

.878695 

571 

283 

.878708 

.121292 

417 

.02 

.999988 

34 

1620 

27 

.895085 

570 

584 

.895099 

.104901 

416 

.02 

.999987 

33 

1680 

28 

.910879 

570 

584 

.910894 

.089106 

416 

.02 

.999986 

32 

1740 

29 

.926119 

570 

585 

.926134 

.073866 

415 

.02 

.999985 

31 

1800 

30 

.940842 

569 

;586 

.940858 

.059142 

414 

.03 

.999983 

30 

I860 

31 

7.955082 

569 

'  587  7.955100 

12.044900 

413 

.02 

9.999982 

29 

1920 

32 

.968870 

569 

587 

.968889 

.031111 

413 

.02 

.999981 

28 

1980 

33 

.982233 

568 

588 

.982253 

.017747 

412 

.02 

.999980 

27 

2040 

34 

7.995198 

568 

589 

7.995219 

12.004781 

411 

.02 

.999979 

26 

2100 

35  8.007787 

567 

590 

8.007809 

11.5)92191 

410 

.03 

.9*9977 

25 

2160 

36   .020021 

567 

:  591 

.020044 

.979956 

409 

.02 

.999976 

24 

2220 

37 

.031919 

566 

592 

.031945 

.968055 

408 

.02 

.999975 

23 

2280 

38 

.043501 

566 

593 

.043527 

.956473 

407 

.03 

.999973 

22 

2340 

39 

.054781 

566 

593 

.054809 

.945191 

407 

.02 

.999972 

21 

2400 

40   .065776 

565 

[594 

.065806 

.934194 

406 

.02 

.999971 

20 

2460  41 

8.076500 

565 

•595 

8.076531 

11.923469 

405 

.03 

9.999969 

19 

2520  42 

.086965 

564 

596 

.086997 

.913003 

404 

.02 

.999968 

18 

2580  1  43 

.097183 

564 

598 

.097217 

.902783 

402 

.03 

.999966 

17 

2640 

44 

.107167 

563 

599 

.107203 

.892797 

401 

.03 

.999964 

16 

2700  45 

.116926 

562 

600 

.116963 

.883037 

400 

.02 

.999963 

15 

2760  46 

.126471  562 

601 

.126510 

.873490 

399 

,03 

.999961 

14 

2820  !  47 

.135810  561 

602 

.135851 

.864149 

398 

.03 

.999959 

13 

2880  48 

.144953 

561 

603 

.144996 

.855004 

397 

.02 

.999958 

12 

2940  49 

.153907 

560 

604 

.153952 

.846048 

396 

.03 

.999956 

11 

3000 

50 

.162681 

560 

605 

.162727 

.837273 

895 

'.03 

.999954 

10 

8060  51 

8.171280 

559 

:607 

8.171328  ! 

11.828672 

393 

.03 

9  999952 

9 

3120  52 

.X79713 

558 

608 

.179763 

.820237 

392 

.03 

.999950 

8 

3180  53 

.187985 

558 

609 

.188036 

.811964 

391 

.03 

.999948 

7 

3240  54 

.196102 

557 

.611 

.196156 

.803844 

389 

.03 

.999946 

6 

3300 

55 

.204070 

556 

612 

.204126 

.795874 

388 

.03 

.999944 

5 

8)60 

56 

.211895 

556 

613 

.211953 

.788047 

387 

.03 

.999942 

4 

3120 

57 

.219581 

555 

615 

.219641 

.780-359 

385 

.03 

.999940 

3 

3  180 

58 

.227134 

554 

1616 

.227195 

.772805 

384 

.03 

.999938 

2 

3)40 

59 

.234557 

554 

618 

.234621 

.765379 

382 

.03 

.999936 

1 

3300 

60 

8.241855 

553 

619 

8.241921 

11.758079 

381   •»» 

9.999934 

0 

4.685 

15.314 

// 

/ 

Cosine. 

q  —  I   Cotang. 

Tang. 

q  +  l 

Dl" 

Sine. 

' 

90° 


89C 


TABLE  XXV. -LOGARITHMIC  SINES, 


178' 


// 

/ 

Sine. 

q-l 

Tang. 

Cotang. 

q  +  l 

Dl" 

Cosine. 

,  i 

!  4.685 

15.314 

3600 

0 

8.241855 

553  1619 

8.241921 

11.758079   381  i 

nn  '  9.999934 

60 

3660 

1 

.249033 

552 

620  !  .249102 

.750898 

380   •"£   .999932 

59 

3720 

2 

.258094 

551 

622   .256165 

.743835 

378  '  'Xo   .999929 

58 

3780 

3 

.263042 

551 

623   .263115 

.736885 

377   'SI  j  .999927 

57 

3840 

4   .269881 

550 

625   .269956 

.730044 

375  ryS;  .999925 

£6 

3900 

5 

.276614 

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11.490800 

274 

•J2  '  «.  9997  74 

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6720 

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.512867 

498  729 

.513098 

.486902 

271 

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.699769 

8 

6780 

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.516726 

497  731 

.516961 

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269 

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6840 

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.999761 

6 

6900 

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494  ;  737 

.  524586 

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263 

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07 

.999757 

5 

6960 

56 

.528102 

492  740 

.528349 

.471651 

260 

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f\Q 

.689753  i  4 

7020 

57 

.531828  491  i  743   .532080 

.467920 

257 

.Ua 
07 

.899748   3 

7080 

58 

.535523  490 

745   .5ao779 

.464221 

255 

.U/ 

(VJ 

.899744   2 

7140 

59 

.539186 

488  ' 

748  !  .539447 

.460553 

252 

.Ui 
Oft 

.999740 

1 

7200  60 

8.542819 

487 

f51\  8.543084   11.456916 

24(1 

•UO 

9.989735 

0 

j 

4.685 

15.314 

! 

,/  j  / 

Cosine.   q  -  I 

Cotang. 

Tang. 

q  +  i\ 

D  r   Sine. 

I 

COSINES,  TANGENTS,   AND  COTANGENTS. 


177° 


I 

' 

Sine. 

D.  r. 

Cosine. 

D.I". 

Tang. 

D.  1". 

Cotang. 

' 

0 

8  542819 

!  9.  999735 

(Yf 

8.543084 

11.456916 

60 

1 

3 
4 
5 

.546422 
.549995 
.553539 
.557054 
.560540 

60*55 

59.07 
58.58 
58.10 
*S7  6.5 

.999731 
.999726 
.999722 
.999717 
.999713 

,\)t 

.08 
.07 
.08 
.07 
O8 

.546691 
.550268 
.553817 
.557336 

.560828 

59!  62 
59.15 
58.65 

58.20 

5rf   170 

.453309 
.449732 
.446183 
.442664 
.439172 

59 
58 
57 
56 
55 

6 

7 
8 
9 
10 

.563999 
.567431 
.570836 
.574214 
.577566 

57^20 
56.75 
56.30 
55.87 
55.43 

.999708 
.999704 
!     .999699 
.999694 
.999689 

.Uo 
.07 
.08 
.08 
.08 
.07 

.564291 
.567727 
.571137 
.574520 

.577877 

i  .  (  X 

57.27 
56.  83 
56.38 
55.95 
55.52 

.435709 
.432273 
.428863 
.425480 
.422123 

54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

8.580892 
.584193 
.587469 
.590721 
.593948 
.597152 
.600332 
.603489 
.606623 

55.02 
54.60 
54.20 
53.78 
53.40 
53.00 
52.62 
52.23 

9.999685 
.999680 
.999675 
.999670 
.999665 
.999660 
.999655 
.999650 
.999645 

.08 
.08 
.08 
.08 
.08 
.08 
.08 
.08 

8.581208 
.584514 
.587795 
.591051 
.594283 
.597492 
.600677 
.603«39 
.606978 

55.10 
54.68 
54.27 
53.87 
53.48 
53.08 
52.70 
52.32 

11.418792 
.415486 
.412205 
.408949 
.405717 
.402508 
.399323 
.396161 
.393022 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

.609734 

51.85 
51.48 

.900640 

.08 
.08 

.610094 

51  .93 
51.58 

.389906 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

8.612823 
.615891 
.618937 
.621962 
.624965 
.627948 
.630911 
.633854 
.636776 
.639680 

51.13 
50.77 
50.42 
50.05 
49.72 
49.38 
49.05 
48.70 
48.40 
48.05 

9.999635 
.999629 
.999624 
.999619 
.999614 
.999608 
.999603 
.999597 
.999592 
.999586 

.10 
.08 
.08 
.08 
.10 
.08 
.10 
.08 
.10 
.08 

8.613189 
.616262 
.619313 
.622343 
.625352 
.628340 
.631308 
.634256 
.637184 
.640093 

51.22 

50.  m 

.  50.50 
50.15 
49.80 
49.47 
49.13 
48.80 
48.48 
48.15 

11.386811 
.383738 
.380687 
.377657 
.374648 
.371660 
.368692 
.365744 
.362816 
.359907 

39 
38 
37 
26 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 

8.642563 
.645428 
.648274 
.651102 
.653911 
.656702 

47.75 
47.43 
47.13 

46.82 
46.52 

9.999581 
|     .999575 
.999570 
.999564 
.999558 
.999553 

.10 
.08 
.10 
.10 
.08 

10 

1  8.642982 
.645853 
.648704 
.651537 
.654352 
.657149 

47.85 
47.52 
47.22 
46.92 
46.62 

11.357018 
.354147 
.351296 
.348463 
.345648 
.342851 

29 
2S 
27 
26 
25 
24 

38 
39 
40 

.659475 
.662230 
.664968 
.667689 

45^92 
45.63 
45.  &5 
45.07 

.999547 
.999541 
.999S&5 
.999529 

•  1  '  ' 

.10 

.10 
.10 

.08 

.659928 
.662689 
.665433 
.668160 

46  .'02 
45.73 
45.45 
45.17 

.340072 
.337311 
.334567 
.331840 

23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 

8.670393 
.673080 
.6757'51 
.678405 
.681043 
.683665 
.686272 

44.78 
44.52 
44.23 
43.97 
43.70 
43.45 

9.999524 
.999518 
.999512 
.999506 
.999500 
.999493 
.999487 

.10 
.10 
.10 
.10 
.12 
.10 

8.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 

44.88 
44.60 
44.35 
44.07 
43.80 
43.53 

11.329130 
.326437 
.323761 
.321100 
.318456 
.315828 
.313216 

19 
18 
17 
16 
15 
14 
13 

48 
49 
50 

.688863 
.6914.8 
.693998 

43.18 
42.92 
42.67 
42.42 

.999481 
.999475 
.999469 

.10 
.10 
.10 
.10 

.689381 
.691963 
.694529 

43'  03 

42.  ¥7 
42.53 

.310619 
.  3080137 
.305471 

12 
11 
10 

51 

52 

8.696543 
.699073 

42.17 
41   Q^ 

9.999463 
.999456 

.12 

1  A 

8.697081 
.699617 

42.27 

11.302919 

.300383 

9 

8 

53 
54 
55 
56 
57 
58 
59 
60 

.701589 
.704090 
.706577 
.709049 
.711507 
.713952 
.716383 
8.718800 

41  .'68 
41.45 
41.20 
40.97 
40.75 
40.52 
40.23 

.999450 
.999443 
.999437 

.999431 
.999424 
.999418 
.999411 
9.999404 

.  lu 

.12 

.10 
.10 
.12 
.10 
.12 
.12 

.702139 
.704646 
.707140 
.709618 
.712083 
.714534 
.716972 
8.719396 

41^78 
41.57 
41.30 
41.08 
40.  &5 
40.63 
40.40 

.297861 
.295354 
.292860 
.290382 
.287917 
.285466 
.283028 
11.280604 

7 
6 
5 
4 
3 
2 
1 
0 

'       Cosine. 

D.  1". 

Sine,     i   D.I'.  1 

Cotang.  1  D.  1".  1     Tang.         ' 

361 


8TC 


TABLE  XXV. -LOGARITHMIC  SINES, 


176° 


' 

Sine. 

D.I". 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

' 

0 

1 

8.718800 
.721204 

40.07 

OQ  OK 

9.999404 
.999398 

.10 

-f  O 

8.719396 

.721806 

40.17 

11.280604 
.278194 

60 

59 

2 
3 
4 
5 

.723595 
.725972 

.728:337 
.730688 

oy  .00 
39.62 
39.42 
39.18 

.999391 
.999384 
.999378 
.999371 

.  i  ~ 

.12 
.10 
.12 

.724204 
.726588 
.728959 
.731317 

39!  78 
39.52 
39.30 

.275796  !  58 
.273412  j  57 
.271041  !  56 
.268683  i  55 

6 

7 
8 
9 

.7313027 
.735354 
.737667 
.739969 

38.98 
38.78 
38.55 
38.37 

oo   17 

.999364 
.999357 
.999350 
.999343 

.12 
.12 
.12 
.12 

.733663 
.735996 
.738317 
.740626 

38^88 
38.68 
38.48 

qo  07 

.266337     54 
.264004     53 
.261683     52 
.259374     51 

10 

.742259 

OO.  1  1 

37.95 

.999336 

!l2 

.742922 

OO./sl 

38.08 

.257078  !  50 

11 
12 
13 
14 
15 
16 
17 
18 

8.744536 
.746802 
.749055 
.751297 
.753528 
.755747 
.757955 
.760151 

37.77 
37.55 
37.37 
37.18 
36.98 
36.80 
36.60 

Q/l        .Q 

9.999329 
.999322 
.999315 
.999308 
.999301 
.999294 
.999287 
.999279 

.12 
.12 
.12 
.12 
.12 
.12 
.13 

8.745207 
.747479 
.749740 
.751989 
.754227 
.756453 
.758668 
.760872 

37.87 
37.68 
37.48 
37.30 
37.10 
36.92 
36.73 

11.254793 
.252521 
.250260 
.248011 
.245773 
.243547 
.241332 

49 
48 

45 
44 
43 
42 

19 
20 

.762337 
.764511 

ob.4o 
36.23 
36.07 

.999272 
.999265 

.12 
.12 

.763065 
.765246 

36.55 
36.35 
3G.18 

'.  236935 
.234754 

41 

40 

21 
22 
23 
24 
25 

8.766675 
•   .768828 
.770970 
.773101 
.775223 

35.88 
35.70 
35.52  ' 
35.37 

9.999257 
.999250 
.999242 
.999235 
.999227 

.12 
.13 
.12 
.13 

8.767417 
.769578 
.771727 
.773866 
.775995 

36.02 
35.82 
35.65 
35.48 

O*"    QO 

11.232583 
.230422 
.228273 
.226134 

.224005 

39 
38 
37 
36 
35 

26 

27 

.777333 
.779434 

35"02 

.999220 
.999212 

.12 
.13 

.778114 
.780222 

OO.6J 

35.13 

.221886 
.219778 

34 
33 

28 
29 
30 

.781524 
.783605 
.785675 

34^68 
34.50 
34.35 

.999205 
.999197 
.999189 

.12 
.13 
.13 
.13 

.782320 
.784408 
.786486 

34  '.80 
34.63 
34.47 

.217680 
.215593 

.213514 

32 
31 
30 

31 

32 
33 

8.787736 
.789787 
.791828 

34.18 
34.02 

QQ     QK 

9.999181 
.999174 
.999166 

.12 
.13 

1  Q 

8.788554 
.790613 
.792662 

34.32 
34.15 

qq   no 

11.211446 

.209387 
.207338 

29 

28 
27 

34 

.793859 

OO.CV) 

QQ    r-A 

.999158 

.lo 

-t  0 

.794701 

oo  .  Uo 

QQ     OQ 

.205299 

26 

35 
36 
37 

.795881 
.797894 
.799897 

OO.  <0 

33.55 
33.38 

.999150 
.999142 
.999134 

.lo 

.13 
.13 

1  Q 

.796731 
.798752 
.800763 

OO.OO 

33.68 
33.52 
qq   q7 

.203269 
.201248 
.199237 

25 
24 
23 

38 
39 

.801892 
.803876 

33!  07 

.999126 
.999118 

•  lo 

.13 

1  Q 

.802765 
.804758 

oo.o* 

as.  22 

qq  r)7 

.197235 
.195242 

22 
21 

40 

.805852 

33!  78 

.999110 

.  lo 

.13 

.806742 

oo.Uf 
32.92 

.193258 

20 

41 

42 
43 

8.807819 
.809777 
.81172(5 

32.63 
32.48 

oo   QK 

9.999102 
.999094 
.999086 

.13 
.13 

1  K. 

8.808717 
.810683 
.812641 

32.77 
32.63 

OO    A*? 

11.191283 
.189317 
.187359 

19 
18 
17 

44 
45 

.813667 
.815599 

32  '.20 

.999077 
.999069 

.lo 
.13 

1  Q 

.814589 
!     .816529 

o£  .Qi 
32.33 

00     OA 

.185411 

.  183471 

16 
15 

46 
47 
48 
49 
50 

.817522 
.819436 
.821343 
.823240 
.825130 

31  '90 
31.78 
31.62 
31.50 
31.35 

.999061 
.999053 
.999044 
.999036 
.999027 

.  lo 
.13 
.15 
.13 
.15 
.13 

.818461 
.820384 
.822298 
.824205 
.826103 

o&  .6\j 

32.05 
31.90 
31.78 
31.63 
31.48 

.181539 
.179616 
.177702 
.175795 

.173897 

14 
13 
12 
11 
10 

51 

52 
53 

54 
55 
56 

57 
58 

8.827011 
.828884 
.830749 
.832607 
.834456 
.836297 
.838130 
.839956 

31.22 
31.08 
30.97 
30.82 
30.68 
30.55 
30.43 

9.999019 
.999010 
.999002 
.998993 
.998984 
.998976 
.998967 
.998958 

.15 
.13 
.15 
.15 
.13 
.15 
.15 

•f  Q 

8.827992 
.829874 
.831748 
.833613 
.835471 
.837321 
.839163 
.840998 

31.37 
31.23 
31.08 
30.97 
30.83 
30.70 
30.58 

QA    A\ 

11.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 
.159002 

9 

8 
7 
6 
5 
4 
3 
2 

59 

.841774 

30.30 

QA    1  Q 

.998950 

.lo 

1  Pi 

.842825 

oU.4  ) 

OA    OO 

.157175 

1 

60 

8.843585 

oU.lo 

9.998941 

.  lo 

8.844644 

OU.O^ 

11.155356 

0 

' 

Cosine. 

D  1". 

Sine. 

D.  r. 

Cotang. 

D.  1".   1     Tang. 

' 

93C 


86° 


COSINES,  TANGENTS,   AND  COTANGENTS. 


175° 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

• 

0 

8.843585 

9.998941 

8.844644 

11.155356 

60 

1 

2 
3 

.845387 
.847183 
.848971 

29.93 

29.80 

.998932 
.998923 
.998914 

.15 
.15 

.846455 
.848260 
.850057 

30.08 
29.95 

.153545 
.151740 
.149943 

59 
58 
57 

4 

.850751 

.998905 

.851846 

.148154 

56 

5 
6 

.852525 

.854291 

29.43 

.998896 
.998887 

.15 

.853628 
.855403 

29.58 

.146372 
.144597 

55 
54 

7 
8 
9 
10 

.856049 
.857801 
.859546 
.861283 

29.20 
29.08 
28.95 
28.85 

.998878 
.998869 
.998860 
.998851 

.15 
.15 
.15 
.17 

.857171 
.858932 
.860686 
.862433 

29.35 
29.23 
29.12 
29.00 

.142829 
.141068 
.139314 
.137567 

53 
52 
51 
50 

11 
12 
13 

8.863014 
.864738 
.866455 

28.73 
28.62 

9.998841 
.998832 
.998823 

.15 
.15 

8.864173 
.865906 
.867632 

28.88 
28.77 

11.135827 
.134094 
.  132368 

49 
48 

47 

14 
15 

.868165 

.869868 

28.38 

.998813 
.998804 

.15 

.869351 
.871064 

28.55 

.130649 
.128936 

46 
45 

16 
17 

18 
19 
20 

.871565 
.873255 
.874938 
.876615 

.878285 

28.17 
28.05 
27.95 
27.83 
27.73 

.998795 
.998785 
.998776 
.998766 
.998757 

.17 
.15 
.17 
.15 
.17 

.87277'0 
.874469 
.876162 
.877849 
.879529 

28.32 
28.22 
28.12 
28.00 
27.88 

.127230 
.125531 
.123838 
.122151 
.120471 

44 
43 
42 
41 
40 

21 

SB 

8.879949 
.881607 

27.63 

9.998747 
.998738 

.15 

8.881202 
.882869 

27.78 

11.118798 
.117131 

39 
38 

23 
24 

25 
26 

27 
28 
29 
30 

.883258 
.884903 
.886542 
.888174 
.889801 
.891421 
.893035 
.894643 

27.42 
27.32 
27.20 
27.12 
27.00 
26.90 
26.80 
26.72 

.998728 
.998718 
.998708 
.998699 
.998689 
.998679 
.998669 
.998659 

.17 
.17 
.15 
.17 
.17 
.17 
.17- 

.ir 

.884530 
.886185 
.887833 
.889476 
.891112 
.892742 
.894366 
.895984 

27.58 
27.47 
27.38 
27.27 
27.17 
27.07 
26.97 
26.87 

.115470 
.113815 
.112167 
.110524 
.108888 
.107258 
.105634 
.104016 

37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
86 

8.896246 
.897842 
.899432 
.901017 
.902596 

26.60 
26.50 
26.42 
26.32 

9.998649 
.998639 
.998629 
.998619 
.998609 

.17 
.17 
.17 
.17 

8.897596 
.899203 
.900803 
.902398 
.903987 

26.78 
26.67 
26.58 
26.48 

11.102404 
.100797 
.099197 
.097602 
.096013 

29 
28 
27 
26 
25 

36 
37 
38 
39 
40 

.904169 
.905736 
.907297 
.908853 
.910404 

26.12 
26.02 
25.93 
25.85 
25.75 

.998599 
.998589 
.998578 
.998568 
.998558 

.17 
.18 
.17 
.17 
.17 

.905570 
.907147 
.908719 
.9102&5 
.911846 

26.28 
26.20 
26.10 
26.02 
25.92 

.094430 
.092853 
.091281 
.089715 
.088154 

24 
23 
22 
21 
20 

41 

42 
43 
44 
45 

8.911949 
.913488 
.915022 
.916550 
.918073 

25.65 
25.57 
25.47 
25.38 

9.998548 
.998537 
.998527 
.998516 
.998506 

.18 
.17 
.18 
.17 

8.913401 
.914951 
.916495 
.918034 
.919568 

25.83 
25.73 
25.63 
25.57 

11.086599 
.085049 
.083505 
.081966 

.080432" 

19 
18 
17 
16 
15 

46 
47 
48 
49 
50 

.919591 
.921103 
.922610 
.924112 
.925609 

25.20 
25.12 
25.03 
24.95 

24  85 

.998495 
.998485 
.998474 
.998464 
.998453 

.17 
.18 
.17 
.18 

.18 

.921096 
.922619 
.924136 
.925649 
.927156 

25.38 
25.28 
25.22 
25.  U 
25.03 

.078904 
.077381 
.075864 
.074351 
.072844 

14 
13 

12 
11 
10 

51 

52 

8.927100 
.928587 

24.78 

9.998442 
.998431 

.18 

8.928658 
.930155 

24.95 

11.071342 

.069845 

9 

8 

53 
54 
55 
56 
57 
58 
59 
60 

.930068 
.931544 
.933015 
.934481 
.935942 
.937398 
.938850 
8.940296 

24.60 
24.52 
24.43 
24.35 
24.27 
24.20 
24.10 

.998421 
.998410 
.998399 
.998388 
.998377 
.998366 
.998355 
9.998344 

.18 
.18 
.18 
.18 
.18 
.18 
.18 

.931647 
.933134 
.934616 
.936093 
.937565 
.939032 
.940494 
8.941952 

24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.30 

.068353 
.066866 
.065384 
.063907 
.062435 
.060968 
.059506 
11.058048 

7 
6 
5 
4 
3 
2 
1 
0 

'       Cosine. 

D.  r. 

Sine. 

D.  1". 

Cotang.     D.  1". 

Tang. 

' 

94° 


333 


85C 


TABLE  XXV. -LOGARITHMIC  SINES, 


174° 


S' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

8.940296 

„,  m       9.998344 

18 

8.941952 

OA    OA 

11.058048     GO 

i 

.941738 

4  "t  '   o           998333 

.  lo 

1  U 

.943404 

^4  .  £\) 
O/l    1Q 

.056590      5!) 

2 

.943174 

oH;"         '.  998322 

.  lo 

1  vl 

.944852 

&*.  lo 

.055148     58 

3 

.944006 

SrS         .998311 

.  lo 

10 

.940295 

24.05 

OQ    OG 

.053705  !  57 

4 

.9460:34 

S2'2x         .998300 

.  lo 

.947734 

/6O.  I/O 
OQ    HA 

.052206     50 

5 

.947456 

SS'iS  :       .998289 

'on 

.949168 

CD.W 

->•*   w*> 

.050832 

55 

G 

.948874 

55  -22  !.     .998277 

.^U       ; 

1  tt 

.950597 

/SO.O/& 

oo  '"o 

.049403 

54 

7 

.950287 

£|-5a         .998266 

.  lo 

.952021 

m,  to 

.047979 

53 

8 

.951696 

SS'S         -998255 

.18 
on     i 

.953441 

28.67 

OQ    KQ 

.040559      52 

9 

.953100 

Sj-£!         .998243 

.•^U 

1  Q 

.954856 

30.  uQ 

.045144      51 

10 

.954499 

i:i  !  •«*»» 

.lo 

.20 

.956267 

23.52 
23.45 

.043733     50 

11 

8.955894 

9o  17      9.998220 

18 

8.957674 

90    0*: 

11.042326     49 

12 

.957284 

oo  iA  l       .998209 

.  lo 

.959075 

V  *  .  OO 

.040925 

48 

13 

.958670 

SS'Jx         .998197 

*1S      i 

.960473 

90*99 

.039527 

47 

14 

.960052 

22  95  ''     -99818G 

.  lo 

.961866 

OQ      - 

.038134 

46 

15 

.961429 

OO    ft1"' 

.998174 

*18 

.963255 

OQ    AI** 

.036745 

45 

16 
17 
18 
19 

20 

.962801 
.904170 
.965534 
.966893 
.968249 

£&  .o< 

22.82 
22.73 
22.65 
22.60 
22.52 

.998163 
.998151 
.998139 
.998128 
.998116 

.  lo 
.20 
.20 
.18 
.20    i 
.20 

.964639 
.966019 
.967394 
.908766 
.970133 

i&j  .  U  i 

23.00 
22.92 

22.87 
22.78 
22.  72 

.035361 
.033981 
.032606 
.081584 

.029867 

44 
43 
42 
41 
40 

21 
22 

8.969600 
.970947 

22.45 

9.998104 

.998092 

.20 

8.G71496 
.972855 

22.65 

11.028504 
.027145 

39 

38 

23 
24 
25 

26 
27 

.972289 
.973628 
.974962 
.976393 
.977619 

22^32 
22.23 
22.18 
22.10 
oo  no 

.998080 
.998068 
.998050 
.998044 
.998032 

!20 
.20 
.20 
.20 

.974209 
.975560 
.976906 

.978248 
.979586 

22.57 
22.52 
22.43 
22.37 
22.30 

oo  OK 

.025791 
.024440 
.023094 
.021752 
.020414 

37 
86 
35 
34 
33 

28 
29 
30 

.978941 
.980259 
.981573 

•at  .  Uo 
21.97 
21.90 
21.83 

.998020 
.998008 
.997996 

!20 
.20 
.20 

.980921 
.982251 
.983577 

3DP*KD 

22.17 
22.10 
22.03 

.019079 
.017749 
.016423 

32 
31 
30 

31 
32 
33 

8.982883 
.984189 
.985491 

21.77 
21.72 

9.997984 
.997972 
.997959 

.20 
.22 

8.984899 
.986217 
.987532 

21.97 

21.92 

11.015101 
.013783 
.012408 

29 

28 
27 

34 

35 
36 
37 
38 
39 
40 

.986789 
.988083 
.989374 
.990660 
.991943 
.993222 
.994497 

21^57 
21.52 
21.43 
21.38 
21.32 
21.25 
21.18 

.997947 
.997935 
.997922 
.997910 
.997897 
.997885 
.997872 

'.20 
.22 
.20 
.22 
.20 
22 
^20 

.988842 
.990149 
.991451 
.992750 
.994045 
.995337 
.996624 

21  '.78 
21.70 
21.65 
21.58 
21.53 
21.45 
21.40 

.011158 
.009851 
.008549 
.007250 
.005955 
.004003 
.003376 

26 
25 
24 
23 
22 
21 
20 

41 

8.995768 

9.997860 

oo 

8.997908 

11.C02092 

19 

42 
43 

.997036 
.998299 

21  .13 
21.05 

.997847 
.997835 

,aS 

.20 
oo 

8.999188 
9.000465 

21  .33 

21.28 

11.000812 
10.999535 

18 
17 

44 

45 
46 
47 

48 

8.999560 
9.000816 
.002039 
.003318 
.004503 

21  .02 
20.93 
20.88 
20.82 
20.75 
on  ^n 

.997822 
.997809 
.997797 
.997784 
.997771 

,309 

.22 
.20 
.22 
.22 

OO 

.001738 
.003007 
.004272 
.005534 
.006792 

21  .22 
21.15 
21.08 
21.03 
20.97 

.998202 
.996993 
.995728 
.944406 
.993208 

16 
15 
14 
18 

12 

49 
50 

.005805 
.007044 

40  .  lO 

20.65 
20.57 

.997758 
.997745 

122 
.22 

.008047 
.009298 

20^85 
20.80 

.991953 

.990702 

11 
10 

51 

9.008278 

9.99773-2 

9.010546 

20  7'3 

10.989454 

9 

52 
53 

.009510 
.010737 

90.45 

.997719 

,     .997700 

!22 
oo 

.011790 
.013031 

20  .'68 

.988210 
.986969 

8 

7 

51 
55 
56 

57 

.011902 
.013182 
.014400 
.015613 

20.42 
20.33 
20.30 
20.22 

.997693 
.997680 
.997667 
.997654 

.let 
.22 
.22 

.22 

.014208 
.015502 
.016732 
.017959 

20^57 
20.50 
20.45 

OA    AC\ 

.985732 
.984498 
.£83208 
.982041 

6 
5 
4 
3 

58 
59 
60 

.016824 
.018031 
9.019235 

20.18 
20.12 
20.07 

.997641 
.997628 
9.997014 

'.22 
.23 

.019183 
.020403 
9.021620 

IcU  .4U 

20.33 
20.28 

.980817 
.979597 
10.978380 

2 
1 
0 

' 

Cosine. 

D.I*. 

Sine. 

D.I". 

Cotang. 

D.  r. 

Tang. 

' 

95' 


6° 


COSINES,  TANGENTS,  AND  COTANGENTS. 


173° 


' 

Sine. 

D.  1".    !   Cosine. 

1 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 

9.019235 
.020435 
.021632 

an  no    :  9.997614 

W.  UU     |        qn-fsni 

10  Q-^        .  yy<oui 
iJ-S         .997588 

.22 

.22 

no 

9.021620 
.022834 
.024044 

20.23 
20.17 
*>n  1  b? 

10.978380 
.977166 
.975956 

60 
59 
58 

3 

4 
5 
6 

7 

.022825 
.024016 
.025203 
.026386 
.027567 

ly.oo 
19.85 
19.78 
19.72 
19.68 

.997-574 

.997561 
.997547 
.997534 
.997520 

,•• 

.22 
.23 
.22 
.23 

.025251 
.026455 
.027655 
.028852 
.030046 

4\)  .  ±4 

20.07 
20.00 
19.95 
19.90 
•in  Q?C 

•    .974749 
.973545 
.972345 
.971148 
.969954 

57 
56 
55 
54 
53 

8 
9 

.028744 
.029918 

19.62 

19.57 

.997507 
.997'493 

.22 
.23 

oo 

.031237 
.032425 

19.  oo 

19.80 

1  (1   '"Q 

.968763 
.967575 

52 
51 

10 

.031089 

19  52 
19.47 

.997480 

rMB 

.23 

.033609 

ly.  <o 
19.70 

.966391 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.032257 
.03:3421 
.034582 
.0:35741 
.036896 
.038048 
.039197 
.040342 
.041485 
.042625 

19.40 
19.35 
19.32 
19.25 
19.20 
19.15 
19.08 
19.05 
19.00 
18.95  , 

9.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 
.997341 

.23 
.22 
.23 
.28 
.23 
.23 
.23 
.23 
.23 
.23 

9.034791 
.035969 
.037144 
.038316 
.039485 
.040651 
.041813 
.042973 
.044130 
.045284 

19.63 
19.58 
19.53 
19.48 
19.43 
19.37 
19.33 
19.28 
19.23 
19.17 

10.965209 
.964031 
.962856 
.961684 
.960515 
.959349 
.958187 
.957027 
.955870 
.954716 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 

9.043762 
.044895 
.046026 

18.88 
18.85 

1  ft   ftO 

9.997327 
.997313 
.997299 

.23 
.23 

9.046434 

.047582 
.048727 

19.13 

19.08 

10.953566 
.952418 
.951273 

39 

38 
37 

24 

.047154 

lo.  0<J 

.997285 

•fif 

.049869 

O 

.950131 

36 

25 

.048279 

18.75 

.997271 

.23 

!     .051008 

18.  9o 

1  Q    (¥% 

.948992 

35 

26 
27 
28 
29 
30 

.049400 
.050519 
.051635 
.052749 
.053859 

18.  68 
18.65 
18.60 
18.57 
18.50 
18.45 

.  99725  r 
.997242 
.997228 
.997214 
.997199 

'.25 
.23 
.23 

.25 
.23 

.052144 
.053277 
.054407 
!     .055535 
i     .056659 

lo.yo 
18.88 
18  83 
18.80 
18.73 
18.70 

.947856 
.946723 
.945593 
.944465 
.943341 

34 
33 
32 
31 
30 

31 

9.054966 

9.997185 

i  9.057781 

1  Q    APi 

10.942219 

29 

32 

.056071 

00~ 

.997170 

..vO 

:     .058900 

lo.oo 
1  o  rn 

.941100 

28 

33 

.057172 

18.  oo 

.997156 

'?? 

1     .060016 

lo.uU 

1  Q    P^1/* 

.939984 

27 

34 
35 
36 
37 

38 

.058271 
.059367 
.060460 
.061551 
.062639 

18  '.27 
18.22 
18.18 
18.13 
1ft  Oft 

.997141 
.997127 
.997112 
.997098 
.997083 

'.23 
.25 
.23 
.25 

.061130 
.062240 
!     .063348 
.064453 
.065556 

lo.ui 

18.50 
18.47 
18.42 
18.38 

1Q    O.> 

.938870 
.937760 
.936652 
.935547 
.934444 

•26 
25 
24 
23 
22 

39 

40 

.063724 
.064806 

lo.  Uo 

18.03 
17.98 

.997068 
.997053 

'.25 

.23 

.066655 
.067752 

lo.o-i 

18.28 
18.25 

.933345 

.932248 

21 

20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.065885 
.066962 
.068036 
.069107 
.070176 
.071242 
.072306 
.073366 
.074424 
.075480 

17.95 
17.90 
17.85 
17.82 
17.77 
17.73 
17.67 
17.  63 
17.60 
17.55 

9.997039 
.997024 
.997009 
.996994 
.996979 
.996964 
.996949 
.996934 
.996919 
.996904 

.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 

9.068846 
.069938 
.071027 
.072113 
.073197 
.074278 
.075356 
.076432 
.077505 
.078576 

18.20 
18.15 
18.10 
18.07 
18.02 
17.97 
17.93 
17.88 
17.85 
17.80 

10.931154 
.930062 
.928973 
.927887 
.926803 
.925722 
.924644 
.923568 
.922495 
.921424 

19 

18 
17 
16 
15 
14 
13 
12 
11 
10 

51 

9.076533 

9.996889 

9.079644 

10.920356 

9 

52 
53 
54 
55 
56 
57 
58 

.077583 
.078631 
.079676 
.080719 
.081759 
.082797 
.083832 

17^47 
17.42 
17.38 
17.33 
17.30 
17.25 
If*  on 

.996874 
.996a58 
1     .996843 
.996828 
.996812 
.996797 
.996782 

'.27 
.25 
.27 
.27 
.25 
.25 

.080710 
.081773 
.082833 
.083891 
.084947 
.086000 
.087050 

17.77 
17.72 
17.67 
17.63 
17.60 
17.55 
17.50 

.919290 
.918227 
.917167 
.916109 
.915053 
.914000 
.912950 

8 
7 
6 
5 
4 
3 
2 

59 
60 

.084864 
9.085894 

i  .M 

17.17 

.996766 
9.996751 

.27 
.25 

.088098 
9.089144 

17.47 
17.43 

.911902 
10.910856 

1 

0 

'       Cosine. 

ix  r. 

Sine. 

D.  1". 

Cotang.     D.  1". 

Tang. 

~ 

96° 


305 


83= 


TABLE  XXV.— LOGARITHMIC  SINES, 


172= 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

•  Tang. 

D.  r. 

Cotang.   ' 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

9.085894 
.080922 
.087947 
.088970 
.089990 
.05)1008 
.092024 
.093037 
.094047 
.095056 
.096062 

17.13 
17.08 

17.05  ; 

17.00  i 
16.97 
16.93 
16.88 
16.83 
16.82 
16.77 
16.72 

9.996751 
.996735 
.996720 
.996704 
.996688 
.996673 
.996657 
.996641 
.996625 
.996610 
.•996594 

.27 
.25 
.27 
.27 
.25 
.27 
.27 
.27 
.25 
.27 
.27 

9.089144 
.090187 
.091228 
.092206 
.093302 
.094336 
.095367 
.090395 
.097422 
.098446 
.099468 

17.38 
17.35 
17.30 
17.27 
17.23 
17.18 
17.13 
17.12 
17.07 
17.03 
16.98 

10.910856 
.909813 
.908712 
.907134 
.900698 
.90C664 
.904033 
.90SC05 
.902518 
.801554 
.900532 

00 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 

9.097065 
.098066 
.099065 
.100002 

16.68 
16.65 
16.62 

•it*  Kiy 

9.996578 
.996562 
.996546 
.996530 

.27 
.27 

'I? 

9.100487 
.101504 
.1C2519 
.103532 

16.95 
16.92 
16.88 

-1  P  CO 

10.899513 

.898496 
.897481 
.896408 

49 
48 
47 
46 

15 
16 
17 
18 
19 
20 

.101056 
.102048 
.103037 
.104025 
.105010 
.105992 

lO.Of 

16.53 
16.48 
16.47 
16.42 
16.37 
16.35 

.996514 
.996498 
.996482 
.996465 
.996449 
.996433 

.%{ 
.27 
.27 
.28 
.27 
.27 
.27 

.104542 
.105550 
.106556 
.107559 
.108560 
.109559 

4O.OO 

16.80 
16.77 
16.72 
16.68 
16.65 
16.62 

.895458 
.894450 
.893444 
.892441 
.891440 
.890441 

45 

44 
43 
42 
41 
40 

21 

9.106973 

1ft  9n 

9.996417 

OQ 

9.110556 

1  A  £Q 

10.889444 

39 

22 
23 
24 
25 
26 
27 
28 

.107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 

10.  *^U 

16.27 
16.23 
16.20 
16.15 
16.12 
16.08 

16  OT 

.990400 
.996384 
.996368 
.990351 
.996335 
.996318 
.996302 

.48 

.27 
.27 
.28 
.27 
.28 
.27 

OQ 

.111551 
.112543 
.113533 
.114521 
.115507 
.116491 
.117472 

lo.oo 
16.53 
16.50 
16.47 
16.43 
16.40 
16.35 

1  t\  *-?*3 

.888449 
.887457 
.886467 
.885479 
.884493 
.888509 
.882528 

38 
37 
36 
35 
34 
33 
32 

29 
30 

.114737 
.115698 

1O.UO 

16.02 
15.97 

.996285 
.996269 

,**O 

.27 
.28 

.118452 
.119429 

10.  oo 

16.28 
16.25 

.881548 
.880571 

31 

SO 

31 

9.116656 

1  CC  flK 

9.996252 

OQ 

9.120404 

1  (*  OO 

10.879596 

29 

32 
33 
34 
35 
36 
37 
38 

.117613 
.118567 
.119519 
.120469 
.121417 
.122362 
.123306 

10.  'JO 

15.90 
15.87 
15.83 
15.80 
15.75 
15.73 

1  H  '""fi 

.996235 
.996219 
.996202 
.996185 
.996168 
.996151 
.996134 

./iO 

.27 
.28 
.28 
.28 
.28 
.28 

OQ 

.121377 
.122348 
.123317 
.124284 
.125249 
.126211 
.127172 

ib.ZZ 
16.18 
16.15 
16.12 
16.08 
16.03 
16.02 

1  K  0*7 

.81-8023 
.877052 
.870083 
.815716 
.874751 
.873189 
.812828 

28 
27 
26 
25 
24 
23 
22 

39 

.124248 

lo  .  <u 

.996117 

.xo 

.128130 

lu.  y  t 

.871810 

21 

40 

.125187 

15.05 
15.63 

.996100 

.28 
.28 

.129087 

15.95 
15.90 

.810913 

20 

41 

9.126125 

1*  KG 

9.99C083 

no 

9.1S0041 

*  1PC  00 

10.6CS959 

19 

42 
43 

.127000 
.127993 

lO.Oo 

15.55 

1  P;  RO 

.996066 
.996049 

./CO 

.28 

OQ 

.130694 
.131944 

10.  oo 

15.83 

1  ^  ft9 

.809006 
.868056 

18 
17 

44 
45 
46 
47 
48 
49 
50 

.128925 
.129854 
.130781 
.131706 
.132680 
.133551 
.134470 

1O.  Do 

15.48 
15.45 
15.42 
15.40 
15.35 
15.32 
15.28 

.996032 
.990015 
.995998 
.995980 
.995963 
.995946 
.995928 

.-CO 

.28 
.28 
.30 
.28 
.28 
.30 
.28 

.132893 
.133839 
.134184 
.135-;  26 
.136667 
.137605 
.138542 

10  .0/& 

15.77 
15.75 
15.70 
15.68 
15.63 
15.62 
15.57 

.867107 
.866161 
.865216 
.864274 
.863333 
.862895 
.861458 

16 
15 
14 
13 
12 
11 
10 

51 
52 
53 

9.135887 
.136303 
.137216 

15.27 

15.22 

9.995911 
.995894 
.995876 

.28 
.30 

9.139476 
.140409 
.141340 

15.55 

15.52 

10.860524  !  9 

.859591  ,  8 
.858660   7 

54 

.138128 

15.20 

!  .995859 

.28 

QA 

.142269 

15.48 

Jr  AE> 

.857131   6 

55 

.139037 

15.15 

.995841 

.oU 

.143196 

u  .  45 

.856804 

5 

56 

.139944 

15.12 

.995823 

.30 

.144121 

15.42 

.855879 

4 

57 

58 

.140850 
.141754 

15.10 

15.07 
i  P;  no 

.995806 
.995788 

.28 
.30 

OQ 

.145044 
.145966 

15.38 
15.37 

-*  pr  QO 

.854956 
.854034 

3 

2 

59 

.142655 

10.U/6 

-1  K  AA 

.995771 

,<;o 

QA 

.146885 

1O.O,4 

.853115 

1 

60 

9.143555 

10.  UU 

9.995753 

.60 

9.147803 

15.30 

10.852197 

0 

/ 

Cosine.   D.  1". 

Sine.    D.  1".  !  Cotang.  D.  1".    Tang. 

' 

97° 


8° 


COSINES,  TANGENTS,  AND  COTANGENTS. 


mc 


' 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.143555 

9.995753 

on 

9.147803 

1  X.    OX 

10.852197 

60 

.  1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.144453 
.145349 
.146243 
.147136 
.148026 
.148U15 
.1498J2 
.150686 
.151569 
.152451 

14.97 
14.93 
14.90 
14.88 
14.83 
14.82 
14.78 
14.73 
14.72 
14.70 
14.65 

.995735 
.995717 
.995699 
.995681 
.995664 
.995646 
.995628 
.995610 
.995591 
.995573 

.oU 
.30 
.30 
.30 
.28 
.30 
.30 
.30 
.32 
.30 
.30 

.148718 
.149632 
.150544 
.151454 
i     .152363 
!     .153269 
.154174 
.155077 
.155978 
.156877 

lo.^o 
15.23 
15.20 
15.17 
15.15 
15.10 
15.08 
15.05 
15.02 
14.98 
14.97 

.851282 
.850368 
.849456 
.848546 
.847637 
.846731 
.845826 
.844923 
.844022 
.843123 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 

9.153330 

1  A    £O 

9.995555 

HI 

9.157775 

~\A   QQ 

10.842225 

49 

12 

.154208 

14.  DO 
MKQ 

.995537 

.ou 

QA 

.158671 

14.  oo 
MQA 

.841329 

48 

13 
14 

.155083 
.  155957 

.58 
14.57  j 

.995519 
.995501 

.oU 
.50 
oo 

.159565 
.160457 

.  yu 

14.87 

1/1    QQ 

.840435 
.839543 

47 
46 

15 

16 
17 
18 
19 
20 

.156830 
.157700 
.158569 
.  159435 
.160301 
.161164 

14.55 
14.50  i 
14.48  1 
14.43 
14.43 
14.38 
14.35 

.995482 
.995464 
.995446 
.995427 
.995409 
.995390 

.oJs 

.CO 
.30 
.32 
.30 
.32 
.30 

.161347 
.162236 
.163123 
.164008 
.164892 
.16577-4 

14.  oo 

14.82 
14.78 
14.75 
14.73 
14.70 
14.67 

.838653 
.837764 
.836877 
.835992 
.835108 
.834226 

45 

44 
43 
42 
41 

40 

21 

9.162025 

-t   4      00 

9.995372 

oo 

9.166654 

1  A    £O 

10.833346 

89 

22 
23 
24 

.1G28S5 
.163743 
.164600 

14.  oo 
14.30 

14.28 

1  A    OQ    i 

.995353 
.995:334 
.995316 

'.32 
.30 

.167532 
.168409 
.169284 

14.  DO 

14.62 
14.58 

•{A     KK 

.832468 
.831591 
.830716 

38 
37 
36 

25 

.165454 

14.  ^O 

.995297 

'oo 

.170157 

14  .OO 

~\A    f;Q 

.829843 

35 

26 

27 
28 
29 

.  166307 
.167159 

.168008 
.168856 

14.  23 

14.20  j 
14.15 
14.13  j 

.995278 
.995260 
.995241 
.995222 

'.30 
.32 
.32 

QO 

.171029 
.171899 
.172767 
.173634 

14.  oo 

14.50 
14.47 
14.45 

.828971 
.828101 
.827233 
.826366 

34 
33 
32 
31 

30 

.169702 

14.10  l 
14.08 

.995203 

.0x5 

.32 

.174499 

14.42 
14.38 

.825501 

30 

31 

9.170547 

1  \  no. 

9.995184 

M 

9.175362 

14  VI 

10.824638 

29 

32 

.171389 

14  .  U-J 
1  -i  n*> 

.995165 

QO 

.176224 

14.  ot 

HQQ 

.823776 

28 

33 

.172230 

1  1  .  04 

.  995146 

*9O 

.177084 

.OO 

-t  A     OA 

.822916 

27 

34  ; 
35 

36 

37  J 

.173070 
.173903 
.174744 
.175578 

14.00 
13.97  i 
13.93 
13.90 

1  O    GG 

.995127 
.995108 
.995089 
.995070 

.o4 

.32 
.32 

.32    : 

QO         i 

.177942 
.178799 
.179655 
.180508 

14.  60 
14.28 
1427 
14.22 
•M  on 

.822058 
.821201 
.820345 
.819492 

26 
25 
24 
23 

38  ! 
39 
40 

.176411 
.  177242 

.178072 

19.  OO    : 

13.85  ; 
13.83 
13.80 

.995051 
.99.5032 
.995013 

.O/C 

.32 
.32 
.33 

.181360 
.182211 
.183059 

14.  *u 

14.18 
14.13 
14.13 

.818640 
.817789 
.816941 

22 
21 
20^ 

41 
42 
43 
41 

45  | 

9.178900 
.179726 
.180551 
.181374 
.182196 

13.77 

13.75    : 

13.72 
13.70  ' 

1  Q    n~f    \ 

9.994993 
.994974 
.994955 
.994935 
.994916 

.32 
32 
.33 
.32 

QO 

9.183907 
.184752 
.185597 
.186439 

.187280 

14.08 
14.08 
14.03 
14.02 

10.816093 
.815248 
.814403 
.813561 
.812720 

19 
18 
17 
16 
15 

46 

.183016 

lo.6<    i 

1  O   £Q 

.9948U6 

.OO 
QO 

.188120 

14.00 

.811880 

14 

47 

.183834 

1  0  .  OO     ; 

.994877 

.0*5 

QO 

.188958 

13.97 

.811042 

13 

48  i 

.184651 

13.62  ! 

-1C     ^O 

.994857 

.OO 

QO 

.189794 

13.93 
10  oo 

.810206 

12 

49 

.185466 

1  o  .  Oo 

10    "  ^ 

.994838 

,4K 

QQ 

.190629 

lo.yy 

HO      QCf 

.809371 

11 

50 

.180280 

lO.O<     ] 

13.53 

.994818 

.00 

.33 

.191462 

lo.oo 
13.87 

.808538 

10 

51 

53 
54 

9.187'092 
.187903 
.  188712 
.189519 

13.52 
13.48 
13.45 

-t  O     AQ 

9.994798 
.994779 
.994759 
.994739 

.32 
.33 
.33 

oo 

9.192294 
.193124 
.193953 
,    .194780 

13.83 
13.82 
13.78 

10.807706 
.806876 
.806047 
.805220 

9 

8 
7 
6 

55 

56 
57 
58 
59 
60 

.190325 
.191130 
.191933 
.1927734 
.193534 
9.194332 

lo.4o  ' 
13.42  ! 
13.38  ! 
13.35 

13.  as 

13.30 

.994720 
.9947-00 
.994680 
.994660 
.994640 
9.994620 

.34 
M    ! 
.33 
.33 
.33 
.33 

.195606 
.196430 
.197253 
.198074 
.198894 
9.199713 

13  .  77 
13.73 
13.72 
13.68 
13.67 
13.65 

.804394 
.803570 
.802747 
.801926 
.801106 
10.800287 

5 
4 
3 

2 
1 
0 

' 

Cosine. 

D.  1".    j 

Sine,     i 

D.I". 

Cotang. 

D.I'. 

Tang. 

' 

367 


81C 


TABLE  XXV.— LOGARITHMIC  SINES, 


1.70C 


' 

Sine. 

D.  1'.       Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang.       ' 

0 

1 

2 
3 
4 
5 

G 
7 
8 
9 
10 

9.194332 
.195129 
.195925 
.196719 
.197511 
.198302 
.199091 
.199879 
.200666 
.201451 
.202234 

13.28 
13.27 
13.23 
13.20 
13.18 
13.15 
13.13 
13.12 
13.08 
13.05 
13.05 

9.994620 
.994600 
.994580 
.994560 
.994540 
.994519 
.994499 
.994479 
.994459 
.994438 
.994418 

.33 
.33 
.33 
.33 
.35 
.33 
.33 
.33 
.35 
.33 
.33 

9.199713 
.200529 
.201345 
.202159 

.202971 
.203782 
.204592 
.205400 
.206207 
;     .207013 
.207817 

13.60 
13.60 
13.57 
13.53 
13.52 
13.50 
13.47 
13.45 
13.43 
13.40 

10.800287 
.799471 
.7'98655 
.797841 
.797029 
.796218 
.795408 
.794600 
.793793 
.792987 
.792183 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 

9.203017 
.203797 
.204577 
.205354 

13.00 
13.00 
12.95 

9.994398 
.994377 
.994357 
.994336 

.35 
.33 
.35 

9.208619 
.209420 
.210220 
.211018 

13.35 
13.33 
13.30 

10.791381 
.790580 

.789780 
.788982 

49 

48 
47 
46 

15 
16 
17 
18 
19 
20 

.206131 
.206906 
.207679 
.208452 
.209222 
.209992 

12.92 
12.88 
12.88 
12.83 
12.83 
12.80 

.994316 
.994295 
.994274 
.994254 
.994233 
.994212 

.35 
.35 
.33 
.35 
.35 
.35 

.211815 
.212611 
.213405 

.214198 
.214989 
.215780 

13.28 
13.27 
13.23 
13.22 
13.18 
13.18 
13.13 

.788185 
.787389 
.786595 
.785802 
.785011 
.784220 

45 
44 
43 
42 
41 
40 

21 
22 

24 
25 
26 

27 
28 
29 

9.210760 
.211526 
.212291 
.213055 
.213818 
.214579 
.215338 
.216097 
.216854 

12.77 
12.75 
12.73 
12.72 
12.68 
12.65 
12.65 
12.62 

9.994191 
.994171 
.994150 
.994129 
.991108 
.994087 
.994066 
.994045 
.994024 

.33 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

9.216568 
.217356 
.218142 
.218926 
.219710 
.220492 
.221272 
.222052 
.222830 

13.13 
13.10 
13.07 
13.07 
13.03 
13.00 
13.00 
12.97 

10.783432 
.782644 
.781858 
.781074 
.780290 
.779508 
.778728 
,777948 
.777170 

39 

38 
37 

3r 

34 
33 
32 
31 

30 

.217609 

12.57 

.994003 

.35 

.223607 

12.92 

.776393 

30 

31 
32 
33 
34 
35 
36 

9.218363 
.219116 
.219868 
.220618 
.221367 
.222115 

12.55 
12.53 
12.50 
12.48 
12.47 
12  43 

9.993982 
.993960 
.993939 
.993918 
.993897 
.993875 

.37 
.35 
.35 
.35 
.37 

OK 

9.224382 
.225156 
.225929 
.226700 
.227471 
.228239 

12.90 

12.88 
12.85 
12.85 
12.80 

10.775618 

.774844 
.774071 
.773300 
.772529 
.771761 

29 

28 
27 
26 
25 
24 

37 
38 
39 

.222861 
.223606 
.224349 
.225092 

12.42 
12.38 
12.38 
12.35 

.993854 
.993832 
.993811 
.893789 

.37 
.35 
.37 
.35 

.229007 
.229773 
.230539 
.231302 

12.77 

12.77 
12.72 
12.72 

.770993 

.770227 
.769461 
.768698 

23 
22 

21 
20 

41 

42 
43 

9.225&S3 
.226573 
.227311 

12.33 
12.30 

9.993768 
.993746 
.993725 

.37 
.35 

9.232065 
.232826 
.233586 

12.68 
12.67 

10.767935 

.767174 
.760414 

19 
18 
17 

44 
45 

.228048 
.228784 

12.27 

.993703 
.993681 

.37 

.234345 
.235103 

12.63 

.765655 
.764897 

16 
15 

46 

47 

.229518 
.230252 

12.23 

.993660 
.993638 

.37 

.235859 
.236614 

12.58 

.764141 
.763386 

34 
13 

48 
49 
50 

.230984 
.231715 
.232444 

12.18 
12.15 
12.13 

.993616 
.993594 
.993572 

.37 
.37 
.37 

.237368 

.238120 
.238872 

12.57 
12.53 
12.53 
12.50 

.762632 
.761880 
.761128 

12 
11 
10 

51 

52 
53 
54 
55 
56 
57 
58 
59 

9.233172 
.233899 
.234625 
.235349 
.236073 
.236795 
.237515 
.238235 
.238953 

12.12 
12.10 
12.07 
12.07 
12.03 
12.00 
12.00 
11.97 

9.993550 
.993528 
.993506 
.993484 
.993462 
.993440 
.993418 
.993396 
.993374 

.37 
.37 

£ 

.37 
.37 
.37 
.37 

9.239622 
.240371 
.241118 
.241865 
.242610 
.243354 
.244097 
.244839 
.245579 

12.48 
12.45 
12.45 
12:42 
12.40 
12.38 
12.37 
12.33 

10.760378 
.759629 
.788882 

.758135 
.757390 
.756646 
.755903 
.755161 
.754421 

9 
8 
7 
6 
5 
4 
3 
2 
1 

60 

9.239670 

9.993351 

9.246319 

10.753681 

0 

' 

Cosine. 

D.  1". 

Sine.       D.  1".   1 

Cotang. 

D.  1'. 

Tang.        ' 

3fiS. 


80° 


10° 


COSINES,  TANGENTS,  AND  COTANGENTS. 


169' 


Sine,     i  D.  1".   |    Cosine.      D.  1".   \     Tang.       D.  1".       Cotang, 


100C 


9.239670 
.2401386 
.241101 
.241814 
.242526 


.248947 

.244656 
.245363 
.246069 
.246775 

9.247478 
.248181 


.849583 


.250980 
.251677 
.252:373 
.253067 
.253761 

9.254453 
.255144 
.255834 
.256523 
.257211 
.257898 
.258583 
.259268 
.2.V.W31 


9.261314 
.261994 


.263351 

.264027 
.264703 
.205377 
.266051 


.2(57395 

9.268065 
.2687:34 
.269402 
.270069 
.270735 
.271400 
.272064 
.272726 
.273388 


9.274708 
.275367 
.276025 
.276681 
.277337 
.277991 
.278645 
.279297 
.279948 

9.280599 


11.93 
11.92 
11.88 
11.87 
11.85 
11.83 
11.82 
11.78 
11.77 
11.77 
11.72 

11.72 

11.70 
11.67 
11.65 
11.63 
11.62 
11.60 
11.57 
11.57 
11.53 

11.52 
11.50 
11.48 
11.47 
11.45 
11.42 
11.42 
11.38 
11.37 
11.35 

11.33 
11.32 
11.30 
11.27 
11.27 
11.23 
11.23 
11.20 
11.20 
11.17 

11.15 
11.13 
11.12 
11.10 
11.08 
11.07 
11.03 
11.03 
11.02 
10.98 

10.98 
10.97 
10.93 
10.93 
10.90 
10.90 
10.87 
10.85 
10.85 


.'.193329 


.998284 


.993240 
.993217 
.993195 
.993172 
.993149 
.993127 


.993036 


.992759 
.992736 
.992713 


.992666 


.992478 


.992190 
9.992166 


.992118 


.991996 

.991971 

9.991947 


.37 


.246319 
.247057 


.248580 

.249264 


.250730  | 

.251461 

.252191 

.252920 

.253648 

.254374 
.255100 
.255824 
.256547 
.257269 
.257990 
.258710 
.259429 
.260146 


9.261578 
.262292 
.263005 
.263717 


.265138 
.265847 
.266555 
.267261 
.267967 

9.268671 
.269375 
.270077 
.270779 
.271479 
.272178 
.272876 
.273573 
.274269 
.274964 

9.275658 
.276351 
.277043 
.277734 

.278424 
.279113 
.279801 
.280488 
.281174 
.231858 

9.282542 

.283225 
.283907 


.285947 


.287301 

.287977 

9.288652 


12.30 
12.28 
12.27 
12.23 
12.23 
12.20 
12.18 
12.17 
12.15 
12.13 
12.10 

12.10 
12.07 
12.05 
12.03 
12.02 
12.00 
11.98 
11.95 
11.95 
11.92 

11.90 
11.88 
11.87 
11.85 
11.83 
11.82 
11.80 
11.77 
11.77 
11.73 

11.73 
11.70 
11.70 
11.67 
11.65 
11.63 
11.62 
11.60 
11.58 
11.57 

11.55 
11.53 
11.52 
11.50 
11.48 
11.47 
11.45 
11.^3 
11.40 
11.40 

11.38 
11.37 
11.35 
11.33 
11.32 
11.28 
11.28 
11.27 
11.25 


10.753681  60 
.752943  59 
.752206  58 
.751470  i  57 
.750736  !  56 
.750002  !  55 
.749270  54 
.748539  I  53 
.747809  52 
.747080  51 
.746352  I  60 

10.745626  i  49 

.744900  48 

.744176  47 

.743453  46 

742731  i  45 
.742010 
.741290 
.740571 

.739854  I  41 

.739137  40 


10.738422 
.737708  j  38 
.736995  I  37 
.736283  36 
.735572 
.734862 
.734153  I  as 
.733445  j  32 
.732739  i  31 
.732033  30 

10.731329  29 
.730625  28 
.729923  27 
.729221  I  26 
.728521 
.727822 
.727124 
.726427 
.725731 
.725036 

10.724342 
.723649 

.722957 

.722266 

.721576 

.720887  i  14 

.720199 

.719512 

.718826 

.718142 

10.717458 
.716775 
.716093 
.715412 
.714732 
.714053 
.713376 
.712699 


10.711348 


Cosine.   I  D.  1". 


Sine.     I  D.I".    1 1  Cotang.  I  D.  1".   !      Tang. 


79° 


TABLE  XXV. -LOGARITHMIC  SINES, 


168° 


1 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.280599 

9.991947 

4O 

9.288652 

10.711348 

CO 

1 

2 
3 
4 
5 

.281248 
.281897 
.282544 
.283190 
.283836 

10.82 
10.78 
10.77 
10.77 

.99192^2 
.991897 
.991W73 
.991848 
.991823 

.42 
.40 
.42 
.42 

.289999 
.290671 
.291312 

.292013 

11.22 
11.20 
11.18 

11.18 

.710674 
.710001 
.709:329 
.708658 
.707987 

59 
58 
57 
56 
55 

6 
7 
8 
9 
10 

.284480 
.285124 
.285T66 

.286408 
.237048 

10.73 
10.70 
10.70 
10.67 
10.67 

.991799 
.991774 
.991749 
.991724 
.991699 

.42 
.42 
.42 
*  .42 
.42 

.292682 
.293350 
.294017 
.294684 
.295349 

11.13 
11.12 
11.12 
11.08 
11.07 

.707318 
.706650 
.705983 
.705316 
.704651 

54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.287688 
.288326 
.288964 
.289600 
.290236 
.290870 
.291504 
.292137 
.292768 
.293399 

10.63 
10.63 
10.60 
10.60 
10.57 
10.57 
10.55 
10.52 
10.52 
10.50 

9.991674 
.991649 
.991624 
.991599 
.991574 
.991549 
.991524 
i     .991498 
I     .991473 
.991448 

.42 

.42 
.42 
.42 
.42 
.42 
.43 
.42 
.42 
.43 

9.296013 
.236677 

.297'3S9 
I     .298001 
.298662 
.299322 
.299980 
.300688 
.301295 
.301951 

11.07 
11.03 
11.03 
11.02 
11.00 
10.97 
10.97 
10.95 
10.93 
10:93 

10.703987 
.703323 
.702661 
.701999 
.701338 
.700678 
.700020 
.699362 
.698705 
.698049 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
23 
27 
28 
29 
30 

9.294029 
.294658 
.295286 
.295913 
.296539 
.297164 
.297788 
.298412 
.299034 
.299655 

10.48 
10.47 
10.45 
10.43 
10.42 
10.40 
10.40 
10.37 
10.35 
10.35 

9.991422 
.991397 
.991372 
!     .991346 
.991321 
1     .991295 
.991270 
.991244 
.991218 
\     .991193 

.42 
.42 
.43 
.42 
.43 
.42 
.43 
.43 
.42 
.43 

9.302607 
.303261 
.303914 
.304567 
.305218 
.305869 
.306519 
.307168 
.307816 
.308463 

10.90 

10.88 
10.88 
10.85 
10.85 
10.83 
10.82 
10.80 
10.78 
10.77 

10.697393 
.096789 

.696086 
.695433 
.6947'82 
.694131 
.693481 
.692832 
.692184 
.691537 

39 
38 
37 
36 
35 
34 
33 
32 
31 
80 

31 

32 
33 
34 

9.300276 
.300895 
.301514 
.302132 

10.32 
10.32 
10.30 

9.991167 
.991141 
.991115 
i     .991090 

.43 
.43 

.42 

9.309109 
.309754 
.310399 
.311042 

10.75 
10.75 
10.72 

10.690891 
.690246 
.689601 
.688958 

29 

28 
27 
26 

35 
36 
37 

.302748 
.303364 
.303979 

10.27 
10.25 

.991064 
!     .991038 
1     .991012 

.43 
.43 

.311685 
.312327 
.312968 

10  .  72 
10.70 
10.68 

.688315 
.687673 
.687082 

25 
24 
23 

38 

.304593 

.99C986 

.313608 

10.67 

.686392 

22 

39 
40 

.305.307 
.3C5819 

10.20 
10.18 

.990960 
!     .990934 

.43 
.43 

.314247 
.314885 

10.63 
10.63 

.6867&S 

.685115 

21 
20 

41 

9.303430 

9.990908 

9.315523 

10.684477 

19 

42 
43 

.307041 
.307650 

10.15 

.990882 
.990855 

.45 

.316159 
.316795 

10.60 

.683841 
.683205 

18 
17 

44 

45 
46 
47 
48 
49 
50 

.308259 
.308867 
.3C9474 
.310080 
.810685 
.311289 
.311893 

10.13 
10.12 
10.10 

10.08 
10.07 
10.07  i 
10.03  i 

.990829 
.990803 
.990777 
.990750 
.990724 
.990697 
.990671 

.43 
.43 
.45 
.43 
.45 
.43 
.43 

.317430 
.318064 
.318697 
.319330 
.319961 
.320592 
.321222 

10.57 
10.55 
10.55 
10.52 
10.52 
10.50 
10.48 

.682570 
.681986 

.(581303 
.680670 
.680039 
.679408 

.678778 

16 
15 
14 
13 
12 
11 
10 

51 

52 

53  i 

9.312495 
.313097 
.313698 

10.08  ! 
10.62  | 

n    OQ    ' 

9.990C45 
.990(518 
.990591 

.45 
.45 

9.321851 
.322479 
.323106 

10.47 
10.45 

10.678140 
.677521 

.676894 

9 
8 

7 

54 
55' 
56 

,314297 
.314897 
.315495 

10.00 
9.97  ! 

.990505 
.9905:38 
.990511 

.45 
.45 

.3237X3 

.324358  i 
.324983 

10.42 
10.42 

.676367 

!  (57501  7  : 

6 
5 
4 

57 

58 

.316092 
.316689 

9.95 

.990485 
.990458 

.45     i 

.325607 
.326231 

10.40 

.674393  i 
.673769  i 

3 
2 

59 
60 

.317284 
9.317879 

9.92 

.990431 
9.990404 

.45 

.326853 

9.327475 

10.37 

.678147 

10.672525 

1 
0 

' 

Cosine. 

D.  1*. 

Sine. 

D.  1". 

Cotang.  i 

D.  1".    1 

Tang.     I 

' 

ior 


370 


73 


COSINES,  TANGENTS,  AND  COTANGENTS. 


167° 


' 

Sine. 

D.  r.  ' 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 

4  • 

9.317879 

.318.173 
.319066 
.319658 
.320249 

9.90 
9.88 
9.87  i 
9.85 

9.990404  i 
.990378 
.990:351 
.990324 

.990297 

.43 
.45 
.45 
.45 

9.327475 
.328095 
.328715 
.  321)334 
.829953 

10.33 
10.33 
10.32 
10.32 

10.673525 

.671905 
.671285 
.670666 
.670047 

60 
59 
58 
57 
56 

s  ; 

G 
7 
8 
9 
10 

.320840 
.321430 
.322019 
.322607 
.323194 
.323780 

9.85 
9.83  i 
9.82 
9.80 
9.78 
9.77 
9.77 

.990270 
.990243 
.990215 
.990188 
.990161 
.9901:34 

!45 

.47 
.45 
.45 
.45 
.45 

.330570 
.331187 
.331803 
.332418 
.333033 
.333646 

I0i28 
10.27 
10.25 
10.25 
10.22 
10.22 

.669430 
.668813 
.668197 
.667582 
.666967 
.666354 

55 
54 
53 

52 
51 
50 

11 

12 
13 
14 
15 

If) 

9.324366 
.324950 
.325534 
.326117 
.326700 
.327281 

9.73 
9.73 

9.72 
9.72 
9.68 

9.990107 
.990079 
.990052 
.990025 
.989997 
.989970 

.47 
.45 
.45 
.47 
.45 

9.334259 
.334871 
.3135482 
.336093 
.336702 
.337311 

10.20 
10.18 
10.18 
10.15 
10.15 
m  IQ 

10.665741 
.665129 
.664518 
.663907 
.663298 
.662689 

49 
48 
47 
46 
45 
44 

17 
18 
19 
20 

.327862 
.328442 
.329021 

.329599 

9.68 
9.67 
9.65 
9.63 
9.62 

.989942 
.989915 
.989887 
:  .989860 

.47 
.45 
.47 
.45 
.47 

.337919 
.338527 
.339ia3 
.339739 

1U.  10 
10.13 
10.10 
10.10 
10.08 

'.661473 
.660867 
.660261 

43 

42 
41 
40 

21 
22 
23 
24 

25 

26 

27 
28 

0.330176 
.•:30753 
.331329 
.331903 
.3324;  8 
.838051 
.383624 
.334195 

9.62 
9.60 
9.57 
9.58 
9.55 
9.55 
9.52 

9.989832 
.989804 
.989777 
.989749 
.9897'21 
.989693 
.989665 
.989637 

.47 
.45 
.47 
.47 
.47 
.47 
.47 

9.340344 
.340948 
.341552 
.342155 
.342757 
.343358 
.343958 
.344558 

10.07 
10.07 
10.05 
10.03 
10.02 
10.00 
10.00 

Q  (J.J 

10.659656 
.659052 
.658448 
.657845 
.657243 
.656642 
.656042 
.655442 

39 
38 
37 
36 
35 
34 
33 
32 

29 
30 

.384767 

.335337 

9^50 
9.48 

.989610 
.989582 

.45 

.47 
.48 

.345157 
.345755 

9!  97 
9.97 

.654843 
.654245 

31 

SO 

31 

9.  £35906 

9AQ 

9.989553 

9.346353 

9qq 

10.653647 

29 

32 

.3.36475 

.4o 

.989525 

'jfo 

.346949 

.yo 

.653051 

28 

33 
34 
35 
36 

.337043 

.3-37610 
.338176 

.338742 

9.47 
9.45 
9.43 
9.43 

.989497 
.989469 
.989441 
.989413 

'.47 
.47 
.47 

.347545 
.348141 
.3-18735 
.349329 

9^93 
9.90 
9.90 

90Q 

.652455 
.651859 
.651265 
.650671 

27 
26 
25 
24 

37 
38 
39 
40 

.339307 
.339871 
.340434 
.340996 

9.42 
9.40 
9.38 
9.37 
9.37 

.989385 
.989356 
.989328 
.989300 

.47 
.48 
.47 
.47 
.48 

.349922 
.350514 
.351106 
.351697 

.CO 

9.87 
9.87 
9.85 
9.83 

.650078 
.649486 
.  .648894 
.648303 

23 

22 
21 
20 

41 

9.341558 

9QK 

9.989271 

9.352287 

9ft*} 

10.647713 

19 

42 

.342119 

.00 

.989243 

.47 

.352876 

.O* 

.647124 

18 

43 

.342679 

9.33 

9QQ 

.989214 

.48 

.353465 

9.  8/2 

90A 

.6465a5 

17 

44 
45 
46 

.343239 
.348797 

.344355 

.00 

9.30 
9.30 

.989186 
.989157 
.989128 

!48 
.48 

.354053 
.  £54(540 
.355227 

.oU 

9.78 
9.78 

9tjf!t 

.645947 
.645360 
.644773 

16 
15 
14 

47 

.344912 

9.28 

.989100 

.47 

.355813 

.  (  i 

9r"r 

.644187 

13 

48 
49 

.845469 

.346024 

9.28 
9.25 

.989071 
.989042 

.48 
.48 

.356398 
.856983 

.  <O 

9.73 
0  ^^ 

.643602 
.643018 

12 
11 

50 

.34(5579 

9.25 
9.25 

.989014 

.47 

.48 

.357566 

9^72 

.642434 

10 

51 

9.347134 

9.988985 

AQ 

i  9.358149 

n  r*n 

10.641851 

9 

52 
53 

.347637 
.348240 

9!  23 

.988956 
.988927 

.4n 
.48 

AQ 

.358731 
.359313 

«!  70 

9/>PV 

.641269 
.640687 

8 

7 

54 
55 
56 

.348792 
.349343 

.349893 

JU8 
9.17 

91  "7 

.988898 
.988869 
.988840 

.48 
.48 
.48 

.859898 

.360474 
.361053 

.Ol 

9.68 
9.  65 

9  AX 

.640107 
.639526 
.6:38947 

6 
5 
4 

57 

.850443 

.  1  ( 

9  15 

.988811 

.48 

.361632 

.uo 

.638368 

3 

58 

.850993 

.988782 

.4o 

.362210 

J.uo 

.637790 

2 

59 

GO 

.351540 

9.,  352088 

9.13 
9.13 

.988753 
9.988724 

.48 
.48 

i  .3627'87 
!  9.368364 

9.62 
9.62 

.637213 
10.636636 

1 
0 

' 

Cosine. 

1  D.  1  . 

Sine, 

D.  r. 

1  Cotang. 

D.  r. 

.  Tang. 

' 

102° 


371 


77° 


13° 


TABLE  XXV.— LOGARITHMIC  SINEP5, 


166° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

t 
' 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 
8 

9.352088 
.352635 
.353181 
.353726 
.354271 
.354815 
.355358 
.355901 
.356443 

9.12 
9.10 
9.08 
9.08 
9.07 
9.05 
9.05 
9.03 
902 

9.988724 

>J88G06 
.988636 
.988007 
.988578 
.988548 
.988519 
.988489 

.48  ! 
.48 
.50  ! 
.48 
.48 
.50 
.48 
.50 

9.363364 
.363940 
.364515 
.365090 
.365664 
.366237 
.366810 
.367882 
.367953 

9.60 
9.58 
9.58 
9.57 
9.55 
9.55 
9.53 
9.52 

10.C86GS6 
.636060 
.085485 
.634910 
.634886 
.633763 
.688190 
.632618 
.632047 

CO 
59 

57 
56 
55 
54 
53 
52 

9 
10 

.356984 
.357524 

9.00 
9.00 

.988460 
.988430 

.50 

.48 

1  .368524 
.369094 

9.50 
9.48 

.631476 
.630906 

51 
CO 

11 
12 
13 

9.358064 
.358603 
.359141 

8.98 
8.97 

9.988401 
.988371 
.988342 

.50 
.48 

9.369663 
.370232 
.370799 

9.48 
9.45 

10.630337 
.629768 
.629201 

49 
48 
47 

14 
15 
16 
17 
18 
19 

.359678 
.360215 
.360752 
.361287 
.361822 
.362356 

8.95 
8.95 
8.92 
8.92 
8.90 

.988312 
.988282 
.988252 
.988223 
.988193 
.988163 

.50 
.50 
.48 
.50 
.50 

.371367 
.371933 
.372499 
.373064 
.373629 
.374193 

9.47 
9.43 
9.43 
9.42 
9.42 
9.40 

.628683 
.628067 
.627301 
.626936 
.626371 
.625807 

46 
45 
44 
43 
42 
41 

20 

.362889 

8.88 

.988133 

.£0 

.374756 

9.38 
9.38 

.625244 

40 

21 
22 
23 
24 
25 

9.363422 
.363954 
.364485 
.365016 
.365546 

8.87 

8.85 
8.85 
8.83 

809 

9.988103 
.988073 
.988043 
.988013 
.987983 

'  .50 
.50 
.50 
.50 

9.375319 
.375881 
.376442 
.377003 
•  .377563 

9.37 
9.35 
9.35 
9.33 

10.624681 
.624119 
.628558 
.622997 
.622437 

£9 

88 
37 
£6 

CO 

2(5 

.366075 

.987953 

.378122 

.621878 

34 

27 

.366604 

870    1 

.987922 

.378681 

9.32 

.621319 

28 
29 

.367131 
.367659 

8.80 

.987892 
.987862 

.50 

.879289 
.379797 

9.30 

.620761 
.620203 

£2 
31 

30 

.368185 

8.77 

.987832 

.50 
.52 

.380354 

9.28 
9.27 

.619646 

oO 

31 

32 

9.368711 
.369236 

8.75 

87* 

9.987801 
.987771 

.50 

i  9.380910 

I  .381466 

9.27 

10.619090 

.618534 

£9 

£8 

33 
34 
35 

.369761 

'.  370808 

8.72 
8.72 

.987740 
.987710 
.987679 

.50 
.52 

.382020 
.382575 
t  .383129 

9.25 
9.23 

.617980 
.617425 
1616871 

26 
25 

36 
37 

.371330 
.371852 

8.70 

.987649 

.987618 

.52 

1  .383682 
.884234 

9.20 

.616318 
.615766 

24 
23 

38 

.372373 

.987588 

i  .384786 

.615214 

22 

39 
40 

.372894 
.373414' 

8.67 
8.65 

.987557 
.987526 

.52 
.50 

,  .385337 

.385888 

9.18 
9.18 
9.17 

.614663 
.614112 

21 
20 

41 
42 
43 
44 

9.373933 
.374452 
.374970 

.375487 

8.65 
8.63 

8.62 

9.987496 
.987465 
.987434 
.987403 

.52 
.52 
.52 

9.386438 
.386987 
•  .387536 

.388084 

9.15 
9.15 
9.13 

10.618562 
.613013 
.612464 
.611916 

19 
18 
17 
16 

45 
46 

.376003 
.376519 

8.60 

.987372 
.987341 

.52 

.888631 
.889178 

9.12 

.611869 
-  .61C822 

15 
14 

47 

.377035 

.987310 

.52 

.389724 

.610276 

13 

48 
49 
50 

.377549 
.378063 
.378577 

8.57 
8.57 
8.53 

.987279 
1  .987248 
!  .987217 

.52 
.52 
.52 

.390270 
.390815 
.391360 

9.08 
9.08 
9.05 

.609730 
.C09185 
.608640 

12 
11 
10 

51 

52 

9-379089 
.379601 

8.53 

9.987186 
.987155 

.52 

1  9.391903 
.392447 

9.07 

10.608097 
.607553 

9 

8 

53 
54 

.380113 
.380624 

8.52 

.987124 

.987092 

.53 

.392989 
.393531 

9.03 

.607011 
.606469 

6 

55 
56 

57 

58 

.381134 
.381643 
.382152 
.382661 

8.48 
8.48 
8.48 
8.45  ! 

.9S7'061 
.987080 
.986998 
.986967 

OCfJCWA 

.52 
.53 
.52 

.52 

|  .394073 
.394614 
.  .395154 
.395694 
qonoqo 

9.02 
9.00 
9.00 
8.98 

.605927 
.C05S86 
.604846 
.604306 
6037'67 

5 

4 

o 

2 

60 

9.383675 

8.45 

9.986904 

.53 

9.396771 

8.97 

10.603229 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

,  Cotang. 

D.  1". 

Tang. 

' 

103° 


872 


76' 


COSINES,   TANGENTS,   AND  COTANGENTS. 


165° 


|i 

I 

' 

Sine. 

D.I". 

Cosine.  !  D.  1".  |  Tang, 

D.  1".   Cotang, 

1 

0 

9.383675 

8AK. 

9.986904 

trt 

9.396771 

8  97 

10.603229 

60 

1 

.384182 

.45 

.986873 

•  O« 
to 

.397309 

8  95 

.602691 

59 

2 

.384687 

8^42 

.986841 

QQAflAn 

.Do 

.53 

.397846 

QQUOQO 

8^95 

4602154 

fllll  (11  ^ 

58 

3 
4 

.385192 

.385697 

8.42 

.yoooUy 
.986778 

.52 

.  oyoooo 
.398919 

8.93 

8AM 

.OU1U1  i 

.601081 

57 

56 

5  !  '.  386201 

8.40 

8f)Q 

.986746 

.53 

to 

.399455 

-Uo 
800 

.600545 

55 

6  1  .386704 
7   .387207 
8   .387709 

.00 
8.38 
8.37 

8  OK 

.986714 
.986683 
.986651 

.00 

.52 
.53 

.399990 
.400524 
.401058 

•  9/9 

8.90 
8.90 

8QQ 

.600010 
.599476 
.598942 

54 
53 

52 

9   .388210 

.o5 

Q  Q'- 

.986619 

.53 

K*> 

.401591 

.00 
800 

.598409 

51 

10   .388711 

o.oo 
8.33 

.986587 

.Oo 

.53 

.402124 

.oO 

8.87 

.597876 

50 

11   9.389211 

800 

9.986555 

to 

9.402656 

80* 

10.597344 

49 

12  j  .389711 
13   .390210 

.00 

8.32 
o  on 

.986523 
.986491 

,OO 

.53 

to 

.403187 
.403718 

.oO 

8.85 

8QK 

.596813 
.596282 

48 

47 

14   .390708 

o.oU 

.986459 

.00 

KQ 

.404249 

.oD 

.595751 

46 

15 

16 
17 

18 
19 
20 

.391206 
.391703 
.392199 
.392695 
.393191 
.393685 

8.30 
8.23 
8.27 
8.27 
8.27 
8.23 
8.23 

.986427 
.986395 
.986363 
.986331 
.986299 
.986266 

.OO 

.53 
.53 
.53 
.53 

.55 
.53 

.404778 
.405308 
.405836 
.406364 
.406892 
.407419 

8^83 
8.80 
8.80 
8.80 
8.78 
8.77 

.595222 
.594692 
.5941(54 
.593636 
.593108 
.592581 

45 
44 
43 
42 
41 
40 

21  I  9-394179 
22  •  .394(373 

8.23 

9.986234 
.986202 

.53 

9.407945 
.408471 

8.77 
8i~- 

10.592055 
.591529 

39 

38 

23 
24 
25 

.395166 
.395658 
.396150 

8.  ',20 
8.20 

8-IQ 

.986169 
.986137 
.986104 

!53 
.55 

KQ 

.408996 
.409521 
.410045 

.  (3 

8.75 
8.73 

8i~q 

.591004 
.590479 
.589955 

37 
36 
35 

26 

.396641 

.  lo 

8-IQ 

.986072 

.DO 

.410569 

.  t'J 

8rv) 

.589431 

34 

27 

.397132 

.Ip 

.986039 

.55 

.411092 

.  (it 
8r*o 

.588908 

33 

28 
29 

.397621 

.398111 

8.15 
8.17 

P»  1  " 

.986007 
.985974 

.53 
.55 

.411615 
.412137 

.  Cii, 

8.70 

8/»0 

.588385 
.587863 

32 

31 

30 

.398UIX) 

8.18 

.985942 

.53 
.55 

.412658 

.00 

8.68 

.587342 

30 

31 

9.3990S8 

9.985909 

9.413179 

10.580821 

29 

32   .399575 
33   .400062 
154   .400549 
35  1  .401035 

8J2 
8.12 
8.10 

8  no 

.985876 
.985843 
.985811 
.985778 

!ss 

.53 

.55 

KK 

.413699 
.414219 
.414738 
.415257 

o.  6* 
8.67 
8.65 
8.65 

8f»n 

.586301 
.585781 
.585262 
.584743 

28 
27 
26 
25 

36   .401520 
37   .402005 
38   .402489 
39   .402972 
40  |  .403455 

.Oo 
8.08 
8.07 
8.05 
8.05 
8.05 

.985745 
.985712 
.985679 
.985646 
.985613 

.55 
.5') 
.55 
.55 
.55 
.55 

.415775 
.416293 
.416810 
.417326 
.417842 

.DO 

8.63 
8.62 
8.60 
8.60 
8.60 

.584225 

.583707 
.583190 
.582674 
.582158 

24 

22 

21 

20 

41 

9.403938 

8  no 

9.985580 

KK 

9.418358 

8  to 

10.581642  19 

42 

.404420 

.Oo 

.985547 

.00 

.418873 

.Oo 
Stfft 

.581127   18 

43 

.404901 

8.02 

.985514 

.55 

.419387 

J)i 
8  toy 

.580613 

17 

44 
45 
46 

47 

.405382 
.405862 
.406341 
.406820 

8^00 
7.98 
7.98 

.985480 
.985447 
-.985414 
.985381 

!55 
.55 
.55 

.419901 
.420415 
.420927 
.421440 

,o< 

8.57 
8.55 
8.55 

8KO 

.580099 
.579585 
.579073 
.578560 

16 
15 
14 
13 

48 
49 

.407299 

.407777 

7.98 
7.97 

7nt 

.985347 
.985314 

.57 
.55 

.421952 
.422463 

.5o 

8.52 

8Kt> 

.578048 
.577537 

12 
11 

CO 

.408254 

.  yo 
7.95 

.985280 

.Of 

.55 

.422974 

.O/w 

8.50 

.577026 

10 

51 

9.408731 

9.985247 

p-r. 

9.423484 

10.576516 

9 

52 
53 

.409207 
.409682 

7.93 
7.93 

.985213 
.985180 

.55 

.423993 

.424503 

8.48 
8.50 

8A<~ 

.576007 
.575-197 

8 

54 

.410157 

7.92 

.985146 

.57 

.425011 

.4< 

8  A** 

.574989 

6 

55 
56 

.410632 
.411106 

7.92 
7.90 

7  DQ 

.985113 
.985079 

.55 
.57 

.425519 
.426027 

.4* 

8.47 

O   A~ 

.574481 
.573973 

5 
4 

57 

.411579 

.  OO 

r*  QQ 

.985045 

.57 

.426534   2'iE 

.573466 

3 

58   .412052 
59   .412524 

4  .00 

7.87 

r-  on* 

.985011 
.9849?'8 

,5t 
.55 

.427041 
.427547 

O.1O 

8.43 

.572959 
.572453 

2 
1 

60  9.412996 

<  .87 

9.984944 

.57 

9.428052 

8.42 

10.571948 

0 

1  1  Cosine.   D.  1'. 

Sine. 

D.  1". 

Cotang.  D.  1". 

Tang. 

' 

101° 


373 


75' 


15° 


TABLE  XXV.— LOGARITHMIC  SINES, 


/ 

Sine. 

D.  1". 

Cosine. 

D.  r.  ! 

Tang. 

D.  1'. 

Cotang. 

/ 

0 

1 

2 

9.412996 
.413467 
.413*8 

7.85 

7.85 

700 

9.984944 
.984910 
.984876 

.57 
.57 

t'V 

9-428052 

.42H55H 
.429062 

8.43 

8.40 

10.571948 

.571442 
.570938 

60 

59 
58 

3 
4 
5 
6 

7 
8 

.414408 
.414878 
.415347 
.415815 
.416283 
.416751 

.00 

7.83 
7.82 
7.80 
7.80 
7.80 

.984842 
.984808 
.984774 
.984740 
.984706 
.984672 

.Di 

.57 
.57 
.57 
.57 
.57 

.429566 
.430070 
.430573 
.431075 
.431577 
.432079 

8^40 
8.38 
8.37 
8.37 
8.37 

.570434 
.5091)80 
.569427 
.568925 
.568423 
.567921 

57 
56 
55 
54 
53 
52 

9 
10 

.417217 
.417684 

7i  78 

7.77 

.984638 
.984603 

.57 
.58 
.57 

.432580 
.433080 

8.35 
8.33 
8.33 

.567420 
.566920 

51 

50 

11 

9.418150 

9.984569 

5>7 

9.433580 

8QO 

10.566420 

49 

12 
13 

.418615 
.419079 

7.75 
7.73 

.984535 
.984500 

i 

.58 

.434080 
.434579 

.OO 

8.32 

8QO 

.565920 
.565421 

48 

47 

14 
15 

.419544 
.420007 

7.75 

7.72  t 

.984466 
.984432 

.'57 

.435078 
.435576 

.04 

8.30 

.564922 
.564424 

46 
45 

16 

.420470 

7.72 

.984397 

.58 

.436073 

8.28 

.563927 

44 

17 

.420933 

7.72 

.984363 

.57 

.436570 

8.28 

.563430 

43 

IS 
19 
20 

.421395 
.421857 
.422318 

7.70 
7.70 
7.68 
7.67 

.984328 
.984294 
.984259 

.58 
.57 
.58 
.58 

.437067 
.437503 
.438059 

8.28 
8.27 
8.27 
8-25 

.562333 

.5(12487 
.561941 

42 
41 
40 

21 
22 

9.422778 
.423238 

7.67 

9.984224 
.984190 

.57 

9.438554 
.489048 

8.23 

10.561446 

.560952 

39 

38 

23 

.423697 

7.65 

.984155 

.58 

RQ 

.439543 

8.25 

.560457 

37 

24 

.424156 

7.65 

.984120 

.OO 
to 

.440036 

8.22 
800 

.559964 

36 

25 
26 
27 
28 
29 
30 

.424615 
.425073 
.425530 
425987 
.426443 
.426899 

7.65 
7.63 
7.62 
7.62 
7.60 
7.60 

.984085 
.984050 
.984015 
.983981 
.983946 
.983911 

.OO 

.58 
.58 
.57 
.58 
.58 

.440529 
.441022 
.441514 
i  .442006 
!  .442497 
.442988 

39 

8.22 
8.20 
8.20 
8.18 
8.18 

.559471 
.558978 
.558486 
.557994 
.557503 
.557012 

85 
34 
33 
32 
31 
30 

7.58 

.60 

8.18 

31 
32 

9.427354 

.427809 

7.58 

9.983875 
.983840 

.58 

KQ 

1  9.443479 
.443968 

8.15 

Ci  1  *7 

10.55G521 
.556032 

29 

28 

&3 

.428263 

7.57 

.983805 

.OO 

to 

i  .444458 

0.  1  t 

.555542 

27 

34 

.428717 

7.57 

.983770 

.OO 

to 

.444947 

!Q  1  Q 

.555053 

26 

35 
36 

.429170 
.429623 

7.55 
7.55 

.983735 
.983700 

.Do 

.-58 

.445435 
.445923 

o.lo 

8.13 

U  1  Q 

.554565 
.554077 

25 

24 

37 

38 

.430075 
.430527 

7.53 
7.53 

.983664 
.983629 

!58 

KQ 

.446411 

.446898 

0.  1*3 

8.12 
81  n 

.553589 
.553102 

23 
22 

39 
40 

.430978 
.431429 

7.52 
7.52 
7.50  • 

.983594 
.983558 

.Of) 

.60 

.58 

.447384 
.447870 

.  lu 

8.10 
8.10 

.552616 
.552130 

21 
20 

41 

9.431879 

ff  KA 

9.983523 

9.448S56 

0  AQ 

10.551P.44 

19 

42 
43 

.432329 

.438778 

*  .5U 
7.48 

.983487 
.983452 

'.58 

.448841 
.449326 

o.  Uo 

8.08 

.551159 
.550674 

IS 
17 

44 
45 

.433226 
.433675 

7.47 
7.48 

.983416 
.988381 

.60 

.58 

.449810 
.450294 

8.07 
8.07 

8A«* 

.550190 
.549706 

16 
15 

46 

47 

.434122 
.434569 

7.45 
7.45 

.983345 
.983309 

.60 
.60 

.450777 
.451260 

.05 

8.05 

8/\K 

.549223 
.548740 

14 
13 

48 

.435016 

7.45 

.983273 

.60 

to 

.451743 

.05 
8  no 

.54K57 

12 

49 

.435462 

7.43 

.983238 

.00 

.452225 

.Uo 

.  5-177  7  5 

11 

50 

.435908 

7.43 

7.42 

.983202 

.60 
.60 

.452706 

8.02 

.547294 

10 

51 
52 
53 
54 

9.436353 
.436798 
.437242 
.437683 

7.42 
7.40 
7.40 

9.983166 

.983130 
.983094 
.983058 

.60 
.60 
.60 

9.453187 
.453668 
.454148 
.454628 

8.02 
8.00 
8.00 

10.546813 

.540332 
.545152 
.545S7'2 

9 

8 
7 
6 

55 

.438129 

7.38 

.983022 

.60 

.455107 

£'Q« 

'.544893 

5 

BO 

.  43857-2 

7.38 

.982986 

.CO 

.455586 

'  ^ 

.544414 

4 

57 

58 

.439014 
.439456 

7.37 
7.37 

.982950 
.982914 

.60 
.60 

.456064 
.456542 

7.9< 
7.97 

7QK 

.548CS6 
.543458 

8 

2 

59 
60 

.439897 
9.440338 

7.35 
7.35 

.982878 
9.982842 

'.GO 

.457019 
9.457496 

.yo 
7.95 

.542981 
10.542504 

1 
0 

' 

Cosine. 

D.  1".  I 

Sine. 

D.  r. 

Cotang. 

D.  1". 

Tang. 

' 

105° 


74° 


374 


16" 


COSINES,  TANGENTS,   AND  COTANGENTS. 


163° 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

/ 

0 

1 

2 
3 

9.440338 

.440778 
.441218 
.441(558 

7.33 
7.33 
7.33 

9.982842 
.982805 
.982769 
!  062733 

.62 
.60 
.60 

f'O 

9.457496 
.457973 
.458449 
.458925 

.35 
.93 
.93 
no 

10.542504 
,542027 
•541551 
.541075 

60 
59 

58 
57 

4 

.4-12096 

7.30 

700 

.982696 

.vz 

f*A 

.459400 

,V5o 

no 

.540600 

56 

5 

.4425:35 

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r"  *>n 

.982660 

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nr\ 

.459875 

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f\f\ 

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55 

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r*  kx> 

.982624 

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.460349 

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.539651 

54 

7 

.443410 

7.  /so 

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.982587 

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OA 

.460823 

.90 
f\f\ 

.539177 

53 

8 

.443847 

i  .  /co 
r*  oo 

.982551 

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no 

;  .461297 

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52 

9 

.444284 

t  .40 

r?  oc* 

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h  .461770 

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50 

11 

9.445155 

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10.587285 

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I  .463186 

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48 

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t  .-,•> 
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.982331 

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r«o 

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.535872 

46 

15 

.446893 

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.982294 

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.464599 

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.535401 

45 

16 

.447326 

7.22 

700 

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00 

.534931 

44 

17 

.447759 

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7  2O 

.982220 

.0-6 
fi2 

.465539 

.00 
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.534461 

43 

18 

.448191 

t  .  ,«t; 

17  OA 

.982183 

,O/5 

(-.) 

.466008 

.O/5 
CO 

.533992 

42 

10 

20 

.448623 
.449054 

/  .<vU 

7.18 

7.18 

.982146 
.982109 

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.62 
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.466477 
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.80 
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.533523 
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41 
40 

21 

9.449485 

rf  1~ 

9.982072 

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9.467413 

78 

10.532587 

39 

22 

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717 

.982035 

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.467880 

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38 

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.450345 

tf  .  1  i 

7  -try 

.981998 

»vS5 

ro 

.468347 

.  to 

r-  o 

.531653 

87 

24 

.450775 

.if 

7  1^ 

.981961 

.0^ 
RO 

.468814 

.  io 

.531186 

36 

25 

26 

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.45]  032 

4  .  1O 

7.13 

7  ic 

.981924 
.981886 

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.63 

fi9 

.469280 
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75 

.530720 
.530254 

35 
34 

27 

.452060 

<  .  lo 
7  13 

.981849 

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AO 

.470211 

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.529789 

33 

28 

.452488 

i  .  lu 

r*  i,) 

.981812 

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AO 

.470676 

.  <O 

.529324 

32 

21) 

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rf  .  1* 
7  12 

.981774 

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no 

.471141 

.  '  -' 

79 

.528859 

31 

80 

.4533-12 

*  .  i  - 

7.10 

.981737 

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.62 

.471605 

.  *o 
.73 

.528395 

80 

31 

9.453768 

7  10 

9.981700 

ftQ 

9.472069 

f«p 

10.527931 

29 

32 

.454194 

J  .  1U 

r*  no 

.981662 

.DO 
<*•> 

.472532 

.  *-* 

r'O 

.527468 

28 

33 

.454(51!) 

I  .  Uo 

7  Oft 

.981625 

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f*O 

.472995 

.  »  -1 

.527005 

27 

31 
35 

.455044 

.4554(5!) 

i  .  Uo 

7.08 

7  07 

.981587 
.981549 

.Do 

.63 

f»O 

.473457 
.473919 

_  . 
.70 
7'0 

.526543 

.526081 

26 
25 

3(5 

.455893 

<  .  VM 

^  0% 

.981512 

«OM 

A*> 

.474381 

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.525619 

24 

37 

.456-316 

f  .UO 
r*  Apr 

.981474 

.OO 

4*0 

.474842 

.Do 

i»D 

.525158 

23 

38 

.456739 

4  .  UO 

r*  A-: 

.981436 

.uo 

RO 

.475303 

.  Oo 

.524697 

22 

39 

4571152 

i  .  UO 

r*  no 

.981399 

»|X0 

/»o 

.475763 

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Off 

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21 

40 

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i  .  (JO 

7.03 

.981361 

.UO 

.63 

.476223 

.  Di 

.67 

.523777 

20 

41 

9.458006 

r*  /v> 

9.981323 

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9.476683 

f*K 

10.523317 

19 

42 

.458427 

<  .UJ£ 
7-  02 

.981285 

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ftQ 

.477142 

.  UO 

65 

.522858 

18 

43 

.458848 

t  .U^ 

r-  AA 

.981247 

.UO 

/>  0 

.477601 

f  S'-l 

.522399 

17 

44 

.459268 

<  .  UU 

r*  AA 

.981209 

.UO 

00 

.478059 

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<»o 

.521941 

16 

45 

.459688 

<  .uu 

7  nn 

.981171 

.UO 

f»Q 

.47'8517 

.Uu 

CO 

.521483 

15 

40 

.460108 

(  .  UU 
6  no 

.981138 

.DO 
/»Q 

.478975 

.  Do 

|*0 

.521025 

14 

47 
48 

.460527 
.460946 

.yo 
6.98 

6(>r/ 

.981095 
.981057 

.OO 

.63 

PQ 

.479432 

.479889 

.0^ 

.62 

f\f\ 

.520568 
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13 

12 

49 

.461364 

.  \n 

6f\ry 

.981019 

.M 

»n 

.48ft345 

.DV 

i*A 

.519655 

11 

5J 

.461782 

.Jt 
6.95 

.980981 

.OO 

.65 

.480801 

.IJU 

.60 

.519199 

10 

51 

9.462199 
.462616 

6,95 

9.980942 
.980904 

.63 

9.481257 
.481712 

.58 

to 

10.5187'43 

.518288 

9 

8 

53 

.468032 

6  .  93 

6  no 

.980866 

.63 

ftp; 

.482167 

.OO 

.517833 

54 
55 

.463448 
.463804 

.  ;;o 

6.93 

.980827 
.980789 

.05 

.63 

.482621 
.483075 

.  5  i 
7.57 

.517379 
.516925 

6 
5 

56 

.464279 

(i  .  92 

/>  rw> 

.980750 

.65 

/»0 

.488529 

7.67 

r»  p;^ 

.516471 

4 

57 

.464094 

o.yji 

.980712 

.DO 
/»K 

.483982 

I  .OO 

r*  tr 

.516018 

3 

58 

.465108 

6.90 
6  on 

.980673 

,o5 

AQ 

.484435 

<  .OU 

7  ^^ 

.515565 

2 

59 

.465522 

.  yu 

GQO 

.980635 

,OO 
at 

.484887 

1  .OO 

r*  t»> 

.515113 

1 

60 

9.465935 

.00 

9.980596 

.UO 

9.485339 

4  .OO 

10.514661 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  1". 

Tang. 

/ 

375 


73° 


TABLE  XXV.— LOGARITHMIC  SINES, 


162° 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1'. 

Cotang. 

' 

0 

9.465935 


9.980596 

9.485339 

10.514661 

60 

1 

2 

.466348 
.466761 

6.88 
6.88 

.980558 
.980519 

'.G5 

StK 

.485791 
.486242 

7.53 
7.52 

.514209 
.513758 

59 

58 

3 
4 
5 

.467173 
.467585 
.467996 

6.87 
6.87 
6.85 

6Q~ 

.980480 
.980442 
.980403 

.DO 

.63 
.65 

AX 

.48(5693 
.487143 
.487593 

7.52 
7.50 
7.50 

r»    RTV 

.513307 
.512857 
.512407 

57 
56 
55 

6 

.46S407 

.OO 

6QO 

.980364 

JDO 

AX 

.488043 

1  .OU 

ry     -o 

.511957 

54 

7 
8 

.468817 
.469227 

.So 
6.83 

Goo 

.980325 

.980286 

.OO 

.65 

AX 

.488492 
.488941 

(  .4o 

7.48 

J7       AQ 

.511508 
.511059 

53 
52 

9 

.469637 

.00 

6QO 

.980247 

.OD 

A" 

.489390 

7'47 

.510610 

51 

10 

.470046 

.OSS 

6.82 

.930208 

.tw 
.65 

.489838 

7i47 

'.510162 

50 

11 

12 

9.470455 
.470863 

6.83 

6QA 

9.930169 
.930130 

.65 

ft" 

9.490286 
.4907*3 

7.45 

10.509714 
.509267 

8 

13 
14 
15 

.471271 
.471679 

.472036 

.OU 

6.83 
6.78 

.930091 
.980352 
.980312 

.DO 

.65 

.67 

.491180 
.491627 
.492073 

7.45 
7.45 
7.43 

.508820 
.508373 
.507927 

2 

45 

16 

.472492 

Grirf 

.979973 

/>•- 

.492519 

"   i  •> 

.507481 

44 

17 

.472398 

.  (  ( 

.979934 

.DO 

.492965 

(  .4o 

.t,07035 

43 

18 
19 
23 

.473304 
.473710 
.474115 

6.77 
6.77 
6.75 
6.73 

.979895 
.979855 
.979816 

.65 
.67 
.65 

.67 

.493410 
.498854 
.494299 

7.42 
7.40 
7.42 
7.40 

.506590 
.506146 
.505701 

42 
41 

40 

21 
22 
23 

9.474519 
.474923 
.475337 

6.73 

6.73 

9.979776 
.979737 
.979697 

.65 
.67 

9.494743 
.495186 
.495633 

7.38 
7.40 

r-    OQ 

10.505257 
.504814 
.504370 

39 

38 

371 

24 

.475730 

6.72 

6*?o 

.979858 

.60 

.496073 

(  .OO 

.503927 

36: 

25 

.476133 

.  16 

.979618 

.67 

.496515 

9& 

.5034a5 

35 

26 

.476536 

6r?{\ 

.979579 

.65 

.496957 

r/or' 

.503043 

34 

27 

.476938 

.  (O 

6ryf\ 

.979539 

.67 

AT 

.497399 

J-W 

.502601 

33 

23 

.477340 

.  t  U 

6AQ 

.979499 

.Of 

Off 

.497841 

r*   O" 

.502159 

32 

29 
30 

.477741 

.478142 

.DO 

6.68 
6.67 

.979459 
.979420 

.o7 
.65 
.67 

.498282 
.498722 

i  .OO 

7.33 
7.35 

.501718 
.501278 

31 
30 

31 
32 
33 

9.478542 
.478942 
.479342 

6.67 
6.67 

9.979380 
.979340 
.979300 

.67 
.67 

9.499163 
.499603 
.500042 

7.33 
7.32 

10.500837 
.500397 
.499958 

29 

28 
27 

34 

.479741 

6.65 

6PX 

.979260 

.67 

.500481 

»*S 

.499519 

26 

35 

.480140 

.00 

.979220 

.67 

.500920 

700 

.499080 

25 

36 

.480539 

M'M 

.979180 

.0< 

.501359 

.o/^ 

rf    QA 

.498641 

24 

37 

.480937 

5'2 

.979140 

.67 

.501797 

(  .60 

.498203 

23 

38 
39 

.481334 
.481731 

D.02 

6.62 

.979100 
.979059 

'.GS 

/>rv 

.502235 
.502672 

7^28 

n    OQ 

.497765 
.497328 

22 

21 

40 

.482128 

6  '.62 

.979019 

.O* 

.67 

.503109 

7.'28 

.496891 

20 

41 
42 
43 
44 

9.482525 
.482921 
.483316 
.483712 

6.60 
6.58 
6.60 

9.978979 
.978939 
.978898 
.978858 

.67 
.68 
.67 

9.503546 
.503982 
.504418 
.504854 

7.27 
7.27 
7.27 

10.496454 
.496018 
.49,5582 
.495146 

18 

s 

45 

46 
47 
48 

.484107 
.484501 
.484895 
.485289 

6.58 
6.57 
6.57 
6.57 

6     PC.** 

.978817 
.978777 
.978737 
.978696 

.68 
.67- 
.67 
.68 

CQ 

.505289 
.505724 
.506159 
.5C6593 

7.25 
7.25 
7.25 
7.23 

7OQ 

.494711 
.494276 
.493841 
.493407 

15 
14 
13 

12 

49 

.485682 

.55 

6KK 

.978655 

.OO 

.507027 

.<o 

.492973 

11 

50 

.436075 

.DO 

6.53 

.978615 

163 

.507460 

^22 

.492540 

10 

51 
52 
53 
54 

9.486467 
.486860 
.487251 
.487643 

6.55 
6.52 

6.53 

9.978574 
.978533 
.978493 
.978452 

.68 
.67 
.68 

/>0 

9.507393 
.508325 
.508759 
.509191 

.22 

.22 
.20 

1  Q 

10.492107 
.491674 
.491241 
.490,809 

9 

8 
7 
6 

55 

.488034 

6.52 
6t/\ 

.978411 

.Go 

CO. 

.509622 

.  lo 

.490378 

5 

56 

.488424 

.DU 

6c-A 

.978370 

.uo 

.510054 

1  Q 

.489946 

4 

57 

.488814 

.50 

Cfc/\ 

973329 

po 

.510485 

.lo 

1ft 

.489515 

3 

58 

.439204 

.DU 

fi  4ft 

.978288 

/>0 

.51C916 

.  lo 
1*7 

.489084 

2 

59 
60 

.489593 
9.489982 

6^48 

!     .978247 
i  9.873206 

.00 
.63 

.511346 
9.511776 

.  1  1 

7.17 

.488654 
10.488224 

1 
0 

' 

Cosine. 

D  1". 

i     Sine. 

D.-r. 

Cotang. 

D.  r. 

Tang. 

' 

107C 


376 


COSINES,  TANGENTS,  AND  COTANGENTS. 


161' 


' 

Sine. 

D.  r.  | 

Cosine. 

D.  r. 

Tang. 

.D.  r. 

Cotang. 

' 

0 

1  1 

9.489982 
.490371 

6.48 

9.978206 
.978165 

.68 

9.511776 
.512206. 

7.17 

10.488224 
.487794 

60 
59 

2 

.45)0759 

6.47  I 

.978124 

AQ 

.512635 

7-|K    i 

.487365 

58 

3 
4 

.491147  I 
.491535  ; 

6.47  i 
6.47 

.978083 
.978042 

.DO 

.68 

/»Q 

.513064 
.513493 

.  !•)    ! 

7.15 

71  ^ 

.486936 
.486507 

57 
56 

5 
6 

.491922 

.492308 

6.45  , 
6.43 

.978001 
.977959 

.UO 

.70 

.513921 
.514349 

.  lo 

7.13 

r-'  1  Q 

.486079 
.485651 

E5 
54 

7  i 

.492695 

6.45 
Gjn 

.977918 

.68 

.514777 

i  .lO 

.485223 

53 

8  ! 
9 

.493081 
.493400 

A6 
6.42 

.977877 
.977835 

!70 

AQ 

.515204 
.515631 

7:i2 

7  10 

.484796 
.484369 

52 
51 

10 

.493S51 

6.42 
6.42 

.977794 

.DO 

.70 

.516057 

i  .  1U   i 

7.12 

.483943 

£0 

11 
13 

9.494236 
.494621 
.495005 

6.42  i 
6.40 

6QQ 

9.977752 
.977711 
.977669 

'  .68 
.70 

9.516484 
.516910 
.517335 

7.10 
7.08 
7  in 

10.483516 
.483090 
.482665 

49 
48 

47 

14 
15 

Hi 

.495388 
.4!  15772 
.496154 

.00 
6.40 
6.37  i 

GQO 

.977628 

.977586 
.977544 

!70 

.70 

Ati 

.517761 
.518186 
.518610 

.  IU 

7.08 
7.07 

r'  CV7 

.482239 
.481814 
.481390 

46 
45 
44 

17 

.496537 

.OO 

.977503 

.OO 

.519034 

i  .  UY 

.480966 

43 

IS 

.496919 

6.37  ! 

.977461 

.70 

.519458 

7  07 

.480542 

42 

19 

.497301 

.  t 

.977419 

"r-A 

.519882 

7ns 

.480118 

41 

20 

.497682 

,  '  ? 

.977377 

.  IV 

.520305 

.Uo 

r*  nr 

.479695 

40 

6.35  i 

.70 

i  .Uo 

21 
22 
23 

9.498064 
.498444 

.498825 

6.33 
6.35  ! 

/>  oo   1 

9.977335 

.977293 
.977251 

.70 
.70 

r*A 

9.520728 
.521151 
.521573 

7.05 
7.03 

r*  rvo 

10.479272 

.478849 

.478427 

39 

38 

37 

24 

.499204 

D  .  O-4 

.977209 

.  it1 

.521995 

t  .Uo 

ri  no 

.478005 

36 

25 

.499584 

6.33 

f*  QO   ' 

.977167 

.70 

.522417 

i  .Uo 

.477583 

35 

26 
27 

.499963 
.500342 

D  .  Op 

6.32 

.977125 

.977083 

.'70 

.522838 
.523259 

7".02 

.477162 
.476741 

34 
33 

28 

.500721 

6.32 

.977041 

.70 

.523680 

7.02 

7  ro 

.476320 

32 

29 

30 

.501099 
.501470 

6.30 
6.28 
6.30 

.976999 
.976957' 

!70 
.72 

.524100 
.524520 

7!oo 

7.00 

.475900 
.475480 

31 
30 

31 

9.501854 

9.976914 

9.524940 

10.475060 

29 

32 

.502231 

6.28 
Goo 

.976872 

.70 

.525359 

fi  QS 

.474641 

28 

33 

.502007 

.^i 

.976830 

.525778 

fi  QQ 

.474222 

27 

34 
35 
36 

.502984 

.503360 
.503735 

6.28 
6.27 
6.25 

.976787 
.976745 
.976702 

'.70 
.72 

.526197 
.520015 
.527033 

6^97 
6.97 

.473803 
.473385 

.472967 

26 
25 
24 

37 

.504110 

6.25 
6  ox 

.976660 

.70 

.527451 

R  OK 

.472549 

23 

38 
39 
40 

.504485 
.504800 
.505234 

,/vO 

6.25 
6.23 

6.23 

.976617 
.970574 
.976532 

!72 
.70 

72 

.527868 
.528285 
.528702 

6  '.95 
6.95 
6.95 

.472132 
.471715 
.471298 

22 
21 
20 

41 
42 
43 

44 

9.505008 
.505981 

!  506727 

6.22 
6.22 
6.22  ! 

9.976489 
.97'644G 
.976404 
.976361 

.72 
.70 
.72 

9.529119 
.529535 
.529951 
^530366 

6.93 

6.93 
6.92 

GOO 

10.470881 
.470405 
.470049 
.469634 

19 

18 
17 
16 

45 
46 

.507099 
.507471 

6^20  j 

GOn 

.976318 
.97627o 

'.72 

.530781 
.531196 

•-•SB 

6.92 
6.  92 

.469219 
.468804 

15 
14 

47 

.507843 

.(*\J    , 

61  Q 

.976232 

°i~O 

.531611 

Con 

.468389 

13 

48 

.50S214 

.  lo 
61Q 

.976189 

°X 

.532025 

.  yu 

Gon  • 

.467975 

12 

49 

.508585 

.lo 

61  fi 

.970140 

-99 

.532439 

.yu  * 

.467561 

11 

50 

.508956 

.  lo 

6.17 

.976103 

!72 

.582853 

6.'  88 

.467147 

10 

51 

9.509326 

9.976060 

9.533266 

COO 

10.466734 

9 

52 

.509696 

6.17 

61  x 

.976017 

.72 
mg 

.533679 

.00 

600 

.466321 

8 

53 
54 

.510065 
.510434 

.  lo 

0.15 

.975974 
.975930 

!73 

.534092 
.534504 

.  Cf  > 

6.87 

.465908 
.465496 

6 

55 

.510803 

6.15 

61  X 

.975887 

.  7  '3 

.534916 

6.87 

.465084 

5 

56 

.511172 

.  lo 

61  Q 

.975844 

*«» 

.535328 

6.87 

.46467'2 

4 

57 
58 
59 

.511540 
.511907 
.512275 

.19 

6.12 
6.13 

6  1O 

.975800 
.975757 
.975714 

!72 
.72 

.535739 
.536150 

.530501 

6.85 
6.85 
6.85 

.464261 
.463850 
.463439 

3 
2 
1 

60 

9.512642 

.1,3 

9.975670 

.73 

9.536972 

6.85 

10.463028 

0 

' 

Cosine. 

1  D.  r. 

Sine. 

D.  i". 

Cotang. 

D.  r. 

Tang. 

' 

108° 


377 


19° 


TABLE  XXV.— LOGARITHMIC  SINES, 


160° 


, 

i 

1 

Sine. 

D.1-.J 

Cosine. 

D.,-. 

Tang. 

D.  r. 

Cotang. 

i 

o 

9.512642 

9.975670 

9.536972 

600 

10.463028 

60 

1 

2 

.513009 
.513375 

6.12 
6.10 

•  .975627 
.975583 

ira 

.537382 
.537792 

-OO 

6.83 

6QQ 

.462618 
.462208 

59 

58 

3 

.513741 

6.10 

.975539 

.73 

.538202 

.00 

.461798 

57 

4 

.514107 

6.  10 

/»  .  O    ' 

.975496 

.72 

m 

.538611 

6.82 

.461389 

£6 

5 

6 
7 
8 
9 
10 

.514472 
.514837 
.515202 
.515566 
.515930 
.516294 

o.to 
6.08 
6.08 
6.07 
6.07  i 
6.07  1 
6.05  ; 

.975452 
.  975408 
.975365 
.975321 
.975277 
.975233 

.  (0 

.73 
.72 
.73 
.73 
.73 
.73 

.539020 
.539429 
.539837 
.540245 
.540653 
.541061 

6^82 
6.80 
6.80 
6.80 
6.80 
6.78 

.400980 
.460571 
.460163 
.459755 

.459347 
.458939 

55 
54 
53 

52 
51 
50 

11 

9.516657 

_ 

9.975189 

9.541468 

10.458532 

49 

12 

.517020 

b.Oo 

6/VJ 

.975145 

.7o  ! 

.541875 

6.78 

.458125 

48 

13 

.517382 

.Uo 

.975101 

.73  i 

.542281 

6.77 

.457719 

47 

14 

.517745 

6.05 

.975057 

.73 

.542688 

6.78 

.457312 

46 

15 

.518107 

6.03 

.975013 

.73 

.543094 

6.77 

6i~e 

.4r.O!X)0 

45 

16 

.518468 

«'A? 

.9749C9 

"fO 

.543499 

.  10 

.466501 

44 

17 

.518829 

o  .  02 

.974925 

.10 

:  .543905 

6.77 

.456095 

43 

18 
19 
20 

.519190 
.519551 
.519911 

6.02 
6.02 
6.00 
6.00 

.974880 
.974836 
.974792 

.75 
.73 
.73 
.73 

.544310 
.544715 
.545119 

6.75 
6.75 
--6  73 
6.75 

.455690 
.455285 

.454881 

42 

41 
40 

21 
22 

9.520271 
.520631 

6.00 

5  no 

9.  974748 
.974703 

.75 

9.545524 

..545928 

6.7'3 

10.454476 

.454072 

39 

38 

23 

.520990 

.Uo 

.974659 

.73 

.546331 

6.72 

.45800'.) 

37 

24 

.521349 

5.98 

.974014 

.75 

.546735 

6.73 

.453265 

36 

25 

.521707 

5.97 

.974570 

.73 

.547'!  38 

6.72 

.452862 

35 

26 
27 

.5220(56 
.522424 

5.98 
5.97 

.974525 

.974481 

.75 
.73 

.547540 
.547943 

6.7'0 
6.7'2 

.452460 
.452057 

34 
33 

28 

.522781 

5.95  i 

.974436 

.75 

.548345 

6.70 

.451055 

32 

29 
30 

.523138 
.523495 

5  .  95 
5.95 

.974391 
.974347 

.75 
.73 

.548747 
.549149 

6.70 
6.7'0 

.451253 

.450851 

31 
30 

5.95 

.75 

6.68 

31 

9.523852 

9.974302 

9.549550 

10.4E0450 

29 

32 

.524208 

5.93 

5  no 

.974257 

.75 

.549951 

6.68 

.450049 

28 

33 

.524564 

.9o 

.974212 

.75 

.550352 

6.08 

.449648 

27 

34 

.524920 

5.93 

.974167 

.75 

.550752 

6.07 

.449248 

26 

35 

.525275 

5.92 

.974122 

.75 

.551153 

6.68 

.448847 

25 

36 

.525630 

5.92 

.974077 

.75 

.551552 

6.65 

.448448 

24 

37 
38 
39 
40 

.525984 
.526a39 
.526(593 
.527046 

5.90 
5.92 
5.90 

5.88 
5.90 

.974032 
.973987 
.973942 
.973897 

.75 
.75 
.75 
.75 
.75 

.551952 
.552351 
.552750 
.553149 

6.67 
6  65 
6.65 
6.65 
6.65 

.448048 
.447649 
!447250 
.446851 

23 
22 
21 

20 

41 
42 
43 
44 
45 

9.527400 
.527753 

.528105 
.528458 
.528810 

5.88 

5.87 
5.88 
5.87 

9.973&52 
.97:3807 
.973761 
.973716 
.973671 

.75 

.77 
.75 
.75 

9.553548 
.553946 
.554344 
.554741 
.555139 

6.63 
6.63 
6.62 

0.03 

10.446452 
.446054 
.445656 
.445259 
.444861 

19 
18 
17 
16 
15 

46 

47 
48 
49 
50 

.529161 
.529513 
.529864 
.530215 
.530565 

5.85 
5.87 
5.85 

5.a5 

5.83 
5.83 

.973625 

.973580 
.973535 
.973489 
.973444 

In 

.75 

.77 
.75 

.77 

.55'  536 
.555933 
.  55(5329 
.556725 
.557121 

0  .  02 
6.62 
6.60 
6.60 
6.60 
6.60 

.444464 

.444067 
.443071 
.443275 

.442879 

14 
13 
12 

11 
10 

51 
52 
53 
54 
55 
56 
57 

9.530915 
.531265 
.531614 
.531963 
.532312 
.532601 
!  .533009 

5.83 
5.82 
5.82 
5.82 
5.82 
5.80 

9.973398 
.973352 
.973307 
.973261 
.973215 
.973169 
.973124 

.77 
.75 

.77 
.77 
.77 
.75 

9.557517 
.557913 
.558308 
.558703 
.559097 
.559491 
.559885 

6.60 
6.58 
6.58 
6.57 
6.57 
6.57 

10.442483 
.442087 
.441692 
.441297 
.440903 
.440509 
.440115 

9 
8 

7' 
6 
5  . 
4 
3 

58 

.53,3357 

5.80 

5r-o 

.973078 

'i~ 

.560279 

6.57 

6p-*-* 

.430721 

2 

59 

.533704 

.  <O 

973032 

•  '  ^ 

.500073 

.01 

.43C327 

1 

CO 

9.534052 

5.80 

9.97298G 

.  1  1 

9.561066 

6.55 

10.438934 

0 

' 

Cosine. 

i  D.  r. 

i  Sine. 

i  D.  r. 

Cotang. 

D.  r. 

Tang. 

' 

109C 


378 


70° 


20° 


COSINES,   TANGENTS,   AND  COTANGENTS. 


159° 


' 

Sine. 

D.  1". 

Cosine. 

D.r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.534052 

{        K    r*o 

9.972986 

9.561066 

" 

10.438934 

60 

1 

2 
3 
4 

1      5 

.534399 
.534745 
.5:35092 
.535438 
!  635783 

O  .  <  0 

5.77 
5.78 
5.77 
5.75 

5r*r- 

.972940 
.972894 
.972848 
.97'2802 
.972755 

'.77 
.77 

.77 
.78 

'     .561459 
.561851 
.562244 
.562636 
.563028 

6!  53 
6.55 
6.53 
6.53 

GRO 

.438541 

.438149 
.437756 
.437364 
.436972 

59 
58  . 
57 
56 
55 

6 

.536129 

.  1  i 

.972709 

'X,  . 

.563419 

.O-«5 

GRO 

.436581 

i  54 

7 
8 
9 
10 

.536474 
.536818 
.537163 
.537507 

5.75 
5.73 
5.75 
5.73 
5.73 

.972663 
.972617 
.972570 
.972524 

.7'  ' 
.78 

r*i  t 

.563811 
.564202 
.564593 
J     .564983 

.  uo 
6.52 
6.52 
6.50 
6.50 

.436180 
.435798 
.435407 
.435017 

i  53 
52 
51 
50 

11 

12 

13 
14 
15 

9.537851 
.538194 
.5:38538 
.5:38880 
.539223 

1     5.72 
i     5.70 
5.70 
5.72 

9.97'2478 
.972431 
.972385 
.972338 
.972291 

.78  ' 
.77 
.78 
.78 

i  9-565373 
.5G57'(»3 
.566153 
.566542 
.566932 

6.50 
6.50 
6.48 
6.50 

6A1? 

10.434627 
.434237 
.433847 
.433458 
.433068 

49 
48 
47 
46 
45 

16 

17 

i     .539565 
.539907 

5.70 
5.70 

.972245 

.972198 

'.78 

.567320 
.567709 

.4* 

6.48 

6AQ 

.43268) 
.432291 

44 
43 

18 

;     .£40249 

5.68 

.972151 

.78 

.568098 

.4o 

.431902 

42 

19 

.540590 

5.68 

.972105 

• 

.568486 

n     A* 

.431514 

41 

20 

.540931 

5.68 
5.68 

.972058 

!78 

.568873 

O.4O 

6.47 

.431127 

40 

21 
22 
23 
24 
25 

:  9.541272 
.541613 
.541953 
.542293 
.542632 

,         5.68 

5.67 
5.67 
5.65 

fr    KK 

9.972011 
.971964 
.971917 
.971870 
.971823 

.78 
.78 
.78 
.78 

9.5G92G1 
.569648 
.570035 
.570422 
.57-0809 

G.45 
G.45 
G.45 
6.45 
G  43 

10.430739 
.430352 
.429965 
.429578 

.429191 

39 

38 
37 
36 

!  35 

26 

.542'.)7'1 

i).  QQ 

-   ... 

.971776 

.571195 

6A9 

.428805 

I  34 

27 
28 

'     .543310 
.543649 

O.OO 

5.65 

.971729 
.971682 

'.78 

r«Q 

.571581 
.571967 

.4o 

6.43 

.428419 
.428033 

33 

32 

29 

.543987 

5H 

.971635 

.  1  O 

170 

.572352 

r    IQ 

.427648 

31 

30 

.544325 

.DO 

!     5.63 

.971588 

.  to 
.80 

.572738 

G.'42 

.427262 

30 

31 
32 
33 

'J.:>4W«:3 
.5450M 
.545338 

5.62 
5.63 

5AA 

9.971540 
.971493 
.971446 

.78 
.78 

QA 

9.573123 
.573507 

.572892 

6.40 
6.42 

10.426877 
.426493 
.426108 

29 

28 
27 

34 
85 

.545674 
.546011 

.oU 
5.62 

5TA 

.971398 
.971351 

.oU 

.78 

QA. 

.574276 
.574660 

G!40 

GA(\ 

.425724 
.425340 

26 
25 

36 
37 

.546347 
.546683 

.OU 

5.60 

.971303 
.971256 

.oU 

.78 

QA 

.575044 
.575427 

,4U 

G.38 

600 

.424956 
.424573 

24 
23 

38 

.547019 

o.60 

.971208 

.OU 

r-o 

.575810 

.  OO 
i\    OQ 

.424190 

22 

39 

.547'354 

5.58 

.971161 

.  10 

QA 

.576193 

0.  oo 

6OQ 

.423807 

21 

40 

.547689 

5.58 
5.58 

.971113 

,oU 

.78 

.576576 

.00 
6.38 

.423424 

20 

41 

9.548024 

9.971066 

QA 

!  9.576959 

vt 

10.423041 

19 

42 
43 

.548:359 
.548603 

5.58 
5.57     j 

.971018 
.970970 

.80 

.80 

QA 

.577341 
.577723 

6^37 

GOK 

.422659 
.422277 

18 

17 

44 

45 

.549027 
.5=19360 

5.57 
5.55     i 

.970922 
.970874 

.oU 

.80 

r*Q 

.578104 
.578486 

.uu 

6.37 

6    QK 

.421896 
.421514 

16 
15 

46 

.549693 

5.55     i 

5tpr 

.970827 

.  to 

QA 

.578867 

.OO 

6     OK 

.421133 

14 

47 
48 

.550026 
.550359 

.OD        i 

5.55     | 

5fr- 

.970779 
.970731 

.oU 

.80 

OA 

.579248 
.57S629 

.CO 

6.35 

600 

.420752 
.420371 

13 
12 

49 
50 

.550692 
.551024 

.OO      ! 

5.53     1 
5.53     ' 

.970683 
.970635 

.OU 

.£0 

.82 

.580009 
.580389 

.00 

6.33 
6.33 

.419991 
.419611 

11 

10 

51 

9.551350 

5.52    ! 

9.970586 

on 

9.580769 

GOQ 

10.419231 

9 

52 

.551687 

f)    fc>> 

.970538 

.oU 

OA 

.581149 

.OO 
COO 

.418851 

8 

53 

54 

.552.  18 
.552349 
.552680 

5.  '53 
5.52 

.970490 
.970^42 
.970394 

.oO 
.80 
.80 

.581528 
.581C07 
.582x86 

.66 

6.32 

6.32 

.418472 
.418093 
.417714 

7 
6 
5 

57 

.553010 
.553341 

5  .50 

5.52 

5AQ 

.970345 
.970297 

.82 
.80 

.582C65 
.583044 

6.32 
6.32 

GOA 

.417335 
.416956 

4 
3 

58 

.553670 

.4o 

.970249 

.80 

.683422 

.OU 

6QA 

.416578 

2 

59 

.554000 

5AO 

.970200 

CA 

.583800 

.oU 

GOG 

.416200 

1 

60 

9.554329 

.4o 

9.970152 

.oU 

9.584177 

.<«u 

10.415823 

0 

' 

Cosine. 

D.r.  1 

Sine,     j 

D.  r.  i 

Cotang. 

D.  r. 

Tang. 

' 

110° 


379 


69= 


TABLE  XXV. -LOGARITHMIC  SINES, 


158° 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.554329 

5^0 

9.970152 

9.584177 

6-0  i 

10.415823 

GO 

1 

.554658 

.4o 

.970103 

.82 

.584555 

U.cU   1 
GftQ 

.415445 

£9 

2 

.554987 

5.48 

.970055 

.80 

.584932 

.A-O 

6  Oft 

.4ir,u(i8 

58 

3 

4 

.555315 
.555643 

5.47 
5.47 

.970006 
.969957 

'.82 

.585309 
.585686 

./SrO 

6.28 

.414691 
.414814 

r  rt 

56 

5 

.555971 

5.47 

.969909 

.80 

.586062 

6.27 

.413938 

55 

6 

7 
8 

.556299 
.556626 
.556953 

5.47 
5.45 
5.45 

.1)69860 
.969811 
.969762 

.82  ' 
.82 
.82 

.580439 
.586815 
.587190 

6.28 
G.27 
6.25 

.413561 
.413185 
.412810 

54 
53 

52 

9 

.557280 

5  .  45 
510 

.969714 

.80  ' 

.587566 

6.27 

.412434 

51 

10 

.  557606 

.43 

5.43 

.969665 

.82 
.82 

.58,941 

6.25 
6.25 

.412059 

CO 

11 

9.557932 

'  9.9C961G 

9.588316 

10.411684 

49 

12 
13 
14 

.558258 
.558583 
.558909 

5.43 
5.42 
5.43 

.969567 
.969518 
.969469 

.82 
.82 
.82 

.588691 
.589066 
-.589440 

G.2o 
6.25 
6.23 

Goo 

.411309 
.410934 
.410560 

48 
47 
46 

15 

.559234 

5.42 

.969420 

.o2  ! 

.589814 

./CO 

.41  01  Hi 

45 

16 

.559558 

5.40 

.969370 

.83 

.590188 

6.23 

Goo 

.409812 

44 

17 

.559883 

5.42 

.969821 

.82  1 

.590562 

.  ^o 

.409488 

43 

18 
19 
20 

.560207 
.560531 
.560855 

5.40 
5.40 
5.40 
5.38 

.969272 
.969223 
.969173 

.82 
.82 
.83 

.82 

.590935 
.591308 
.591681 

6.22 
6.22 

6.^2. 
6.22 

.406065 

.40£fi!)2 
.408319 

42 
41 
40 

21 
22 

9.561178 
.561501 

5.38 

9.C69124 
.969075 

.82 

9.592054 
.592426 

6.20 

10.407946 
.407574 

89 

38 

23 
24 
25 

.561824 
.562146 

.562468 

5.38 
5.37 
5.37 

.969025 
.968976 
.966926 

.83 
.82 
.83 

.5027!)!) 
.593171 
.593542 

6.22 
6.20 
6.18 

.407201 
.406829 
.406458 

37 
86 
35 

26 

.562790 

5.37 

5nrt 

.968877 

.82 

no 

.593914 

61ft 

.406086 

34 

27 
28 
29 
30 

.563112 
.563433 
.563755 
.564075 

.of 
5.35 
5.37 

5.  as 

5.35 

.968827 
.968777 
.968728 
.968678 

.OO 

.83 

.82 
.83 
.83 

.594285 
.594656 
.595027 
.595398 

.  lo 

6.18 
6.18 
6.18 
G.17 

.405715 
.405344 
.404973 
.404602 

33 
:  32 
81 

i  £0 

31 

9.564396 

5OQ 

!  9.968628 

QO 

9.595768 

61" 

10.404232 

I  09 

32 

.564716 

.OO 

:  .968578 

.OO 

.596138 

.  i  , 

01  ^ 

.403862 

'  28 

33 

.565036 

5.33" 

.968528 

.83 

.596508 

.1  < 

6ir» 

.40341)2 

;  27 

34 

.565356 

5.33 

5na 

.968479 

.82 

.59687'8 

.!< 

61  K 

.403122 

26 

.565676 

.OO 

.968429 

.83 

.507247 

.15 

.402753 

25 

36 
37 
38 
39 
40 

.565995 
.566314 
.566632 
.566951 
.567269 

5  .  32 
5.32 
5.30 
5.32 
5.30 
5.30 

.968379 
.968329 
i  .968278 
i  .968228 
.968178 

.83. 
.83 
.85 
.83 
.83 
.83 

.597'616 
.597985 
.598354 
.598722 
.599C91 

0.15 
6.15 
6.15 
6.13 
6.15 
6.13 

.402384 
.402015 
.401646 
.401278 
.400909 

!  24 
23 
22 
21 
20 

41 

9.567587 

**  fW> 

9.968128 

CO 

9.599459 

6-(O 

10.400541 

If, 

42 
43 
44 

.567'904 
.568222 
.568539 

o.Jto 
5.30 
5.28 

.968078 
.968027 
i  .967977 

.00 

.85 
'.83 

.599827 
.600194 
.600562 

.  lo 
6.12 
6.13 

61  O 

.400178 
.399806 
.399438 

1! 

;  17 

1C 

45 

46 

47 

.568856 
.569172 
.569488 

5.28 
5.27 
5.27 

.967927 
.967876 
.967826 

.83 
.85 
.83 

.600929 
.601296 
.601663 

.!« 
6.12 
6.12 

U~  1fl 

.399071 
.398704 
.398337 

16 
!  14 
13 

48- 
49 
50 

.569804 
.570120 
.570435 

5.27 
5.27 
5.25 
5.27 

:  .967775 
.SG7725 
.007074 

!83 
.85 

.83 

.602029 
.C02395 
.6C2761 

.  lu 
6.10 
6.10 
6.10 

.397'971 
.397605 
.397239 

1  - 
11 
10 

51 
52 

9.570751 
.571066 

5.25. 

9.9G7G24 
.1)67573 

.85 

9.603-127 
.603493 

6.10 

p  no 

10.3S6873 
.396507 

9 

8 

53 

54 

.571380 
.571695 

5.23 
5.25 

5OQ 

.967522 
i  .967471 

.85 
.85 

QQ 

.603858 
.604223 

().0o 

6.08 

(•  (\Q 

.396142 
.395777 

i  7' 
6 

55 
56 
57 
58 
59 
60 

.57'2009 
.572323 
.572636 
.572950 
.573263 
9.573575 

.60 

5.23 
5.22 
5.23 
5.22 
5.20 

.967421 
;  .967370 
!  .967319 
.967268 
!  .967217 
|  9.967166 

.CO 

.85 
.85 
.85 
.85 
.85 

.604588 
.6C4053 
.GC5317 
.605682 
.606046 
9.606410 

o.uo 
6.08 
6.07 
6.08 
6.07 
6.07 

.395412 
.395047 
.394683 
.394318 
.393954 
10.393590 

5 
4 

|  1 

!  o 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

;  Cotang. 

D.  1". 

Tang. 

' 

iir 


COSINES,  TANGENTS,  AND  COTANGENTS. 


157< 


'    Sine. 

D.  r. 

Cosine.   D.  r.  j  Tang. 

D.  1". 

Cotang. 

, 

|j  • 

0 
1 
2 
3 
4 

9.573575 

.573888 
.574200 
.574512 
.574824 

5.22 

5.20 
5.20 
5.20  i 

9.967166 
.967115 
.967064 
.967013 
.966961 

.85 
.85 
.85 
.67 

9.606410 
.606773 
.607137 
.607500 
.607863 

6.05 
6.07 
6.C'5 
6.05 

6  no 

10.393580 
.893227 
.8C2663 
.892800 

.392137 

60 
59 

£8 
57 
56 

5 
6 

.57'5136 
.575447 

5;18 

5-4  Q 

.966910 
.966859 

'.S>  I 

Off 

.608225 
.608588 

.Uo 

6.05 
6  no 

.891775 
.391412 

55 
54 

8 

.575758 
.576069 

.lo 
5.18 

.066808 
.966756 

•  CO 

.67  ! 

OK 

.608950 
.609312 

.Do 
6.03 

.891050 

.390688 

53 
52 

9 

.576379 

5.17 

.966705 

.CO   ! 

.609674 

6.03 

.81)0326 

51 

10 

.57UC89 

5.17  j 
5.17  i 

.966653 

.87  i 

.85   : 

.610036 

6.03 
6.02 

.889964 

50 

11 

9.576999 

5-ity 

9.966602 

9.610397 

6  no 

10.S89C03 

49 

12 
13 

.577809 

.577618 

.li 

5.15  I 

51  %. 

.966550 
.966499 

!85  ! 

or* 

.610759 
.611120 

.Uo 

6.02 
6  fin 

.889241 

.8686'80 

48 

47 

14 

.577927 

.10 

.966447 

.or 

.611480 

.uu 

.888520 

46 

15   .578286 

5.15 

.966395 

.87 

.611841 

6.02 
6nn 

.888159 

45 

16   .578545 

5.15 

.966344 

.65  j 

.612201 

.00 

.887799 

44 

17 

.578853 

5.  13 

.966S92 

.87  ! 

.612561   °-}j" 

.387439 

43 

18 

.579162 

5.15 

5-1  Q 

.966240 

•g   !  .612921 

O.UU 

.887079 

42 

19    579470 

.lo 

.966188 

•?i  i   .613281 

6.1  0 

.386719 

41 

20 

.579777 

5.12 
5.13 

.866136 

!8o    -613641 

6.00 
5.98 

.886859 

40 

21 

9.580085 

51  O 

9.966085 

en* 

9.614000   ,  QQ 

10.386000 

89 

22  1  .580392 

.  1/& 

.S66033 

.o7 

.614359   °'no 

.385641 

38 

23  i  .580699 

5.12 
5  in 

.965981 

•jg    .614718   g  -Eg 

.885282 

37 

24 
25 

.581005 
.581312 

.  JU 
5.12 

.965929 
.965876 

!68 

.615077 
.615435 

U.  tJO 

5.97 

.384923 
.384565 

36 

85 

26 
27 

29 

.581618 
.581924 
.582229 
.582535 

5.  10 
5.10 
5.08 
5.10 
K  nfi 

.965824 
.965772 
.965720 
.965668 

.87  ' 
.87 
.87 
.87 

CO 

.615793 
.616151 
.616509 
.616867 

5  .  97 
5.9/ 
5.97 
5.97 

e  nt 

.884207 
.883849 
.383491 
.383133 

34 
33 
32 
31 

30 

.582840 

o.Uo 
5.08 

.965615 

.CO 

.87 

.617224 

o.yo 
5.97 

.382776 

30 

31 
32 

9.583145 
.583149 

5.07 

f  AQ 

9.965563 
.965511 

.67 

9.617582 
.617939 

5.95 
5  no 

10.382418  !  28 
.382061  28 

33 
34 

36 
37 
88 

39 
40 

.583754 

.584058 
.584361 
.584665 

.584968 
.585272 
.585574 
.585877 

o.Oo 
5.07 
5.05 
5.07 
5.05  i 
5.07 
5.03 
5.05 
"5.03 

.965458 
.965406 
.965353 
.965301 
.965248 
.965195 
.965143 
.965090 

.88 
.87 
.88 
.87 
.88  ! 
.88 
.87 
.88 
.88 

.618295 
.918652 
.619008 
.619364 
.619720 
.620076 
.620432 
.620787 

.yo 
5.95 
5.93 
5.93 
5.93 
5.93 
5.93 
5.92 
5.92 

.381705 
.881348 
.380992 
.380636 
.880280 
.379924 
.378E68 
.379213 

27 
26 
25 
24 
23 
22 
21 
20 

41  !  9.580179 

9.965037 

oo 

9.621142 

10.378858  !  19 

42    .586482 

*'no 

.964984    -}5 

.621497 

5.92 

5  no 

.378503 

18 

43  !   .586783 

K  f>Q 

.964931  i   ;g2 

.621852 

.MB 

.378148 

17 

44  i  .587085 

o.Oo 

.964879  i   -E 

.622207 

5.92 

.377793 

16 

45  '  .587386 

5.02 

.964826  I   -SS 

.622561 

5.  90 

.377439 

15 

46 

47 

48 

.587688 
.587989 
.688289 

5.03 
5.02 
5.00 

.964773 
.964720 
.964666 

.00 

.88 
.90 

.622915 
.623269 
.623623 

5.90 
5.90 
5.90 

.377085 
.376731 

.376377 

14 
13 
12 

49 

.58S5JK) 

5.02  < 

.964613    -58 

.623976 

5.8? 

.376024 

11 

50 

.588890 

.964560 

.624330 

5.90 

.375670 

10 

5  .  UO  ;            .88 

5.88 

51 

52 
53 

9.589190 

.589489 
.589789 

4.98 
5.00 

9.964507 
.964454 
.964400 

.88 
.90 

OQ 

9.624683 
.625036 

.625388 

5.88 

5.87 
500 

10.375317 

.374664 
.374612 

9 
8 

54 

.590088 

4  .  Jo  i 

.964347 

.CO 

.625741 

.00 

.374259 

6 

55 

.590387 

4.  J8 

.964294 

.88  | 

.626093 

5.87 

.373907 

5 

66 

.590686 

40" 

.964240 

.90  1 

oo 

.626445 

5.87 

.373555 

4 

57 

58 

.590984 
.591282 

4.  '97 
A  07 

.964187 
.964133 

.CO 

.90 

QQ 

.626797 
.627149 

5  .  87 
5.87 

5  or* 

.373203 
.372851 

3 
2 

59 
60 

.591580 
9.591878 

4.  yV 

4.97 

.064080 
9.964026 

.CO 

.90 

.627501 
9.627852 

.Ol 

5.85 

.372499 
10.372148 

1 

0 

'  \  Cosine,  i  D.  1".  ||  Sine.   D.  1".  |  Cotang.  |  D.  1". 

Tang.    ' 

67' 


23° 


TABLE  XXV. -LOGARITHMIC  SINES, 


156° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1".  j 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

9.591878 
.592170 

4.97 

4  OX 

9.964020 
.963972 

.90 

00 

9.627K52 
.628203 

5.85 

10.372148 
.371797 

60 
59 

2 

.592473 

iva 

4Q~ 

.963919 

.TO 

on  ; 

.628554 

5.85 

5  Off 

.371446 

58 

3 
4 
5 

.592770 
.593357 
.593303 

.  «7i) 

4.95 
4.93 

A  QO 

.963805 
.983811 
.963757 

.yu 
.90  i 
.90 

go 

.628905 
.629255 

.029800 

.00 
5.83 

5.85 

5QQ 

.371095 
.370745 
.370394 

57' 
56 
55 

6 
7 
8 
9 
10 

.593653 
.593955 
.534251 
.594547 
.594342 

4.  \i<i 

4.93 
4.93 
4.95 
4.92 
4.92 

.983704 
.963050 
.963593 
.903542 
.933483 

.03 

.90 
.90 
.93 
.90 
.90 

.629950 
.630300 
.030650 
.031005 
.G31355 

.OO 

5.83 

5.83 
5.82 
5.83 
5.83 

.370044 
.869694 
.809844 
.808995 
.808645 

54 
53 

.-,2 
51 
£0 

11 
12 
13 

9.535137 
.595432 
.595727 

4.92 

4.92 

4  O1 

9.903434 
.983373 
.963325 

.92  i 
.93 

no 

9.031704 
.032053 
.632402 

5.82 
5.82 

10.368290 
.307947 
.367598 

49 
48 
47 

14 
15 

.590321 
.590315 

.  y  j 
4.93 

4O'l 

.983271 
.963217 

.yj 

.90 
an 

.632750 
.033099 

5.80 
5.82 

.307250 
.300901 

40 
45 

16 
17 
18 
19 

20 

.536C33 
.598903 
.597193 
.597490 

.597783 

.  y  j 

4.93 

4.83 

4.90  ; 

4.88 
4.87 

.983163 
.963108 
.963054 
.932993 
.962945 

.yo. 
.92 
.90 
.92 
.90 
.92 

.633447 
.633795 
.634143 

.634490 
.634838 

5.80 
5.80 
5.80 
5.78 
5^80 
5.78 

.300553 
.30(5205 
.365857 
.365510 
.365162 

44 
43 
42 

41 

40 

21 
22 
23 
24 

9.593075 
.598383 
.593360 
.593352 

4.83  : 
.87 
.87 

Q*f 

9.962390 
.962836 
.962781 
.962727 

.90 
.92 
.90 

9.635185 
.635532 
.635879 
.636220 

5.78 

5.78 
5.78 

10.364815 
.304408 
.364121 
.363774 

39 
38 
37 
30 

25 

.599244 

i  .87 

Q7 

.902672 

.92 

oo 

.636572 

5.77 

.363428 

35 

26 
27 

.599536 
.593327 

.Of 

.85- 
ox 

.962617 
.962502 

,vz 
.92 

(\f\ 

.630919 
.637265 

5.78 
5.77 

!  883081 

.3G2T35 

34 
33 

23 

.633118 

1  .00 

Q~ 

.962503 

.yu 

.637611 

5.77 

.302389 

32 

23 
33 

.603403 

.600703 

'  .  O.J 

.85 
.83 

.98.2453 

.962333 

.92 
.92 
.92 

.637956 
.638302 

5.75 
5.77 
5.75 

.362044 
.301098 

31 

30 

31 
32 
33 

9.603930 
.601233 
.601570 

4.83 

4.83 

4QQ 

9.962343 
.962283 
.902233 

.92 
.92 

9.638047 
.6:38932 
.639337 

5.75 
5.75 

10.31)1353 
.301008 
.360003 

29 
28 
27 

34 
35 

.601800 
.002150 

.00 
4.83 

4QO 

.90217'8 
.962123 

.92 
.92 

oo 

.639682 
.640027 

5.75 
5.75 

.360318 
.859973 

20 
25 

33 
37 

.602433 

.602723 

.04 

4.82 

4QO 

.962067 
.902012 

.Uo 

.92 

no 

.640371 
.640716 

5.73 
5.75 

5M 

.359G29 
.859384 

24 
23 

33 

.603017 

.2H 

4  on   i 

.961957 

.y* 

.641000 

.  I-' 

.358940 

22 

39 
40 

.603305 
.603594 

.o(J 
4.82   ; 

4.89 

.961902 
.961846 

.92 
.93 

.92 

.641404 
.641747 

5.73 
5.72 
5.73 

.858590 
.358253 

21 

20 

41 
42 

9.603882 
.604170 

4.80 

4*7*2   ' 

9.961791 
.961735 

.93 

no 

9.642091 
.642434 

5.72 

10.357909 

.357500 

19 

18 

43 

.604457 

.  to   | 

4-  QA 

.931080 

.y* 

.642777 

5.72 

.357223 

17 

44 

.604745 

:.OU 

4ryo 

.9616.24 

.93 

no 

.643120 

5.72 

.356880 

16 

45 

.605032 

.  to 
A  70 

.961569 

,y.« 

QO 

.643463 

5.72 

5rfi) 

.350537 

15 

43 

.605319 

4.  to 
4r*o 

.961513 

.  yo 

oo 

.643800 

.  fX 

5r'i\ 

.a56194 

14 

47 
43 
49 

.605806 
.605892 
.606179 

.  to 

4.77 
4.78 

.961458 
.981402 
.901340 

.V6 

.93 
.93 

'  no  1 

.644148 
.644490 

.044832 

.  (U 

5.70 
5.70 

.355852 
.a55510 

.355108 

13 

1.' 
11 

50 

.608405 

4.77 
4.77 

.901230 

.yo 
.92 

.645174 

5.70 
5.70 

.354820 

10 

51 
53 
53 

54 

9.605751 
.607036 
.607322 
.607607 

4.75 

4.77 
4.75 

9.901235 
.901179 
.901123 
.961007 

.93 
.93 
.93 

9.645516 
.645857 
.648193 
.646540 

5.G8 
5.70 

5.08 

10.354484 
.354143 

.353801 
.353460 

9 
6 

0 

55 
53 
57 
53 
53 

.607892 
.608177 
.603461 
.608745 
.609029 

4.75 
4.75 
4.73 
4.73 
4.73 

4'*'O 

.901011 
.900955 
.960399 
.980843 
.930788 

.93 
.93 
.93 
.93 
.95  ' 

OO   ; 

.646881 
.647222 
.647562 
.647903 
.648243 

5.68 
5.68 
5.67 
5.68 
5.67 

.a53119 
.352778 
.353438 
.352097 
.351757 

5 
4 
3 
2 
1 

60 

9.603313 

.  i  O 

9.960730 

.JO   ; 

9.048583 

5.67 

10.351417 

0 

1 

Cosine. 

D.  r.  j 

Sine. 

D.  1". 

Cotang. 

D.  r. 

Tang. 

' 

113= 


332 


66° 


24° 


COSINES,  TANGENTS,  AND  COTANGENTS. 


155° 


> 

Sine. 

D.I". 

Cosine. 

D.  1". 

Tang. 

D.  1".  i 

Cotang. 

' 

0 

9.609313 

473  | 

9.960730 

no 

9.648583 

fc  P*?   i 

10.351417 

CO 

I 

3 
4 
5 

.609597 
.609880 
.610104 
.610447 
.610729 

4!72 
4.73  ! 
4.72 
4.70 

.960674 
.960618 
.960561 
.960505 
.960448 

.Uo 
.93 

.95  : 
.93 
.95 

OQ   s 

.648923 
.649263 
.649602 
.649942 
.650281 

O.U«   1 

5.67  1 
5.65  i 
5.67 
5.65 

K  «*r 

.351077 
.350737 
.350398 
•   .350058 
.349719 

59 
58 
57 
56 
65 

6 

7 

.611012 
.611294 

4  ""2  '• 
4.70 
4  70  ' 

.9(50392 
.960385 

.UtJ    ! 

.95  | 

.650620 
.650959 

O.OO 

5.65 

.349380 
.349041 

54 

53 

8 

.611576 

4  Ma 

.960279 

°a~ 

.651297 

5ftX 

.348703 

52 

9 
10 

.611858 

.612140 

.  i\J 

4.70 
4.68 

.960222 
.960165 

!95 
.93 

.651636 
.651974 

.DO 

5.63 
5.63 

.348364 
.348026 

51 

50 

11 

9.612421 

PQ 

9.960109 

9.652312 

5f»n 

10.347688 

49 

.612702 

'  .OO 

.960052 

q? 

.652650 

.63 
5AQ 

.347350 

48 

13 

.612983 

>   AQ 

.959995 

Q- 

.652988 

.Do 

.347012 

47 

14 

.0132(54 

.DO 
!oo 

.959938 

no 

.653326 

5  .  63 

.346674 

46 

15 

.01:3545 

.DO   ; 
.  a~ 

.959882 

QX 

.653663 

5.02 

t  /»rt 

.346.337 

45 

16 

.613825 

*  .Of 

prt   ] 

.959825 

.yo 

.654000 

O.O4 

.346000 

44 

17 

.614105 

.or 

.959768 

"on 

.654337 

5.62 

.345663 

43 

18 

.614385 

4/»r* 

.959711 

.  yo 

n~ 

.654674 

5.62 

.345326 

42 

19 

.614(565 

.Ol 

.959654 

.yo 

9** 

.655011 

5.62 

.344989 

41 

20 

.614944 

4.  DO 

4.65 

.959596 

i 
.95 

.655348 

5.62 
5.60 

.344652 

40 

21 
22 
23 

24 
25 

26 

9.615223 
.615502 
.615781 
.616060 
.616338 
.616610 

4.65 
4.65 
4.65 
4.63 
4.63 

4  OB 

9.959539 
.959482 
.959425 
.959368 
.959310 
.959253 

.95 
.95 
.95 
.97 
.95 

9.655684 
.656020 
.656356 
.656692 
i  .657028 
!  .6573(54 

5.60 
5.60 
5.60 
5.60 
5.60 

K  to 

10.344316 
.343980 
.343644 
.343308 
.342972 
.342636 

39 
38 
37 

3G. 
34 

27 
28 
29 
30 

.616894 
.617172 
.617450 

.017727 

.bo 

4.63 
4.63 
4.62 
4.02 

.959195 
.959138 
.959080 
.959023 

!95 
.97 
.95 

.97 

.657699 
.658034 
1  .658309 
.658704 

O.OO 

5.58 
5.58 
5.58 
5.58 

.342301 
.341966 
.341631 
.341296 

33 
32 
31 
30 

31 

9.618004 

9.958965 

9.659039 

10.340961 

29 

32 
33 
34 

.618&81 
.618558 

.618834 

4!  08 

4.60 

.958908 
.958850 
.958792 

.97 
.97 

.659373 
.659708 
.660042 

5^58 
5.57 

.340627 
.340292 
.£39958 

28 
27 
26 

35 

.619110 

4.60 

.9587:34 

.97 

.660376 

5.57 

.339624 

25 

36 
37 

.619386 
.61960.2 

4.60 
4.60 

.958677 
.958619 

.95 
.97 

.660710 
.661043 

5.57 
5.55 

.339290 
.338957 

24 
23 

38 
39 
40 

.619938 
.620213 

.620488 

4.60 
4.58 
4.58 
4.58 

.958561 
.958503 
,  '.958445 

.97 
.97 
.97 
.97 

.661377 
.661710 
.662043 

5.57 
5.55 
5.55 
5.55 

.338623 

.338290 
.337957 

22 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.620763 
.621038 
.621313 
.621587 
.621861 
.622135 
.622409 
.622682 
.622956 
.623229 

4.58 
4.58 
4.57 
4.57 
4.57 
4.57 
4.55 
4.57 
4.55 
4.55 

1  9.958387 
.958329 
.958271 
.958213 
.958154 
.958096 
|  .958038 
!  .957979 
i  .957921 
.957863 

.97 
.97 
.97 
.98 
.97 
.97 
.98 
.97 
.97 
.98 

9.662376 
.662709 
.668042 
.608375 

.663707 
i  .664039 
i  .664371 
:  .664703 
i  .665035 
.665366 

5.55 
5.55 
5.55 
5.53 
5.53 
5.53 
5.53 
5.53 
5.52 
5.53 

10.337624 
7337291 
.336958 
.336625 
.366293 
.335961 
.335629 
.335297 
.834965 
.3346:34 

19 
18 
17 
10 
15 
11 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 

9.623502 
.623774 
.624047 
.624:319 
.624591 
.624863 
.6251:35 

4.53 
4.55 
4.53 
4.53 
4.53 
4.53 

4K9 

i  9.957804 
.957746 
.957687 
.957628 
.957570 
.957..  11 
.957452 

.97 
.98 
.98 
.97 
.98 
.98 

1  9.665698 
1  .666029 
.666360 
.666691 
.667('21 
.667352 
.667682 

5.52 
5.52 
5.52 
5.50 
5.52 
5.50 

10.334302 

.ami 

.333640 
.333309 
.332979 
.332648 
.832318 

9 

8 
7 
6 
5 
4 
3 

58 

.625406 

,095 

.957393 

*Q7 

.668013 

5.52 

5trk 

.331987 

0 

59 

.625677 

4  CO 

.957335 

'  a  i 

.668843 

.uU 

.881657 

1 

60 

9.625948 

tCSf 

9.957276 

, 

9.668673 

5.50 

10.33132> 

0 

' 

Cosine. 

D.  1".  : 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

'  • 

383 


65° 


25° 


TABLE  XXV.— LOGARITHMIC  SINES, 


154° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

j  Tang. 

D.  1'. 

Cotang. 

' 

0 

9.625948 

9.957276 

9.668673   ,  df   10.331327 

60 

1 

.626219 

i'ro 

.9572W 

.»o 

no 

.669002   °'*°     .330998 

59 

2 

.626190   *"£ 

.957158 

.yo 

no 

.669332 

rS  !   .330668 

!  58 

3 

.G2o7(iO   J-92 

.957099 

.'Jo 
nfl 

.669681 

grn     .330339 

57 

4 
5 

.627030   J'°J 
.  62730  J   fJS 

.957040 
.956981 

.Uo 

.98 

Inn 

.669001 

.670:320 

5.50 

5.48 

.330009 
.829(580 

!  56 
55 

6 

.627570  j  J-|2 

.956921 

.UU 

MS 

.670649 

5.48 

.329351 

54 

7 

.627840   *•;£ 

.9568(52 

.  \to 

f\O 

.670977 

5.47 

.329023 

53 

8 

.628109   2'*g 

.956803    -JS 

.671306 

5.48 

54Q 

.328694 

i  52 

9 
10 

.62&S78 
.628647 

4.48 
4.48 

.9.56741 
.956684 

1.00 
.98 

.671635 
.671963 

Ao 
5.47 
5.47 

.328365 
.328037 

51 
50 

11 
13 
13 
14 

15 
16 
17 
18 
19 
20 

9.628916 
.629185 
.629453 
.629721 
.629989 
.6:30257 
.630524 
.630792 
.631059 
.631326 

4.48 
4.47 
4.47 
4.47 
4.47 
4.45 
4.47 
4.45 
4.45 
4.45 

9.956625 
.956566 
.956506 
.956447 
.956:387 
.956327 
.956268 
.956208 
.956148 
.956089 

.98 
1.00 
.98 
1.00 
1.00 
.98 
1.00 
1.00 
.98 
1.00 

9.672291 
.672619 
.672947 
.673274 
.673602 
.673929 
.674257 
.674584 
.674911 
.675237 

5.47 
5.47 
5.45 
5.47 
5.45 
5.47 
5.45 
5.45 
5.43 
5.45 

10.327709 
.327381 
.327053 
.326726 
.326398 
.326071 
.325743 
.325416 
.325089 
.324763 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 

22 

9.631593 
.631859 

4.43 

9.956029 
.955969 

1.00 

1/Vl 

9.675564 
.675890 

.  40  j  10.324436 
g'J  i   .324110 

39 

38 

23   .632125 

4.4<3 

.955909 

.UU 

.676217 

g'Jg     .323783 

37 

24   .632392 

4.45 

.955849 

1  .00 

.676543 

36 

25  1  .632658 

4  Aft 

.955789 

1.00 

1AA 

.676869 

M2     .'323131 

35 

26   .632923 
27   .633189 
28  1  .633454 

.4% 
4.43 
4.42 

.955729 
.955669 
.955609 

.00 
1.00 
1.00 

no 

.677194   °'^ 
.677520   5-43 
.677846   MS 

.322806 
.322480 
.322154 

34 
88 
32 

29   .633719 
30  i  .633984 

4^42 
4.42 

.955548 
.955488 

.Uo 
1.00 
1.00 

.678171 
.678496 

5.42 
5.42 

.321829 
.321504 

31 
30 

31 
32 

9.634249 
.634514 

4.42 

9.955428 
.955368 

1.00 
Ino 

9.678821 
.679146 

5.42 

5iO 

10.321179 

.320854 

29 
28 

33 
34 
35 
36 

.634778 
.635042 
.635306 
.635570 

4^40 
4.40 
4.40 

4Af\ 

.955307 
.955247 
.955186 
.955126 

.Ui 
1.00 
1.02 
1.00 

1AO 

.679471 
.679795 
.680120 
.680444 

.4.J 
5.40 
5.42 
5.40 

5A(\ 

.320529 
.320205 
.319880 
.319556 

27 
26 
25 
24 

37 

qb 

.635834 

,4U 

4.38 

.955065 

.(j% 
1.00 

.680768 

.4U 

5.40 

.319232 
o-t  contt 

23 

.).) 

OO 

39 

!  636360 

4  38 

40Q 

.'954944 

1.02 

1681416 

5.40 

.  o  J  oyUo 
.318584 

AH 

21 

40 

.636623 

.00 

4.38 

.954883 

i!oo 

.681740 

5.40 
5.38 

.318260 

20 

41 
42 

9.636886 
.637*48 

4.37 

4QQ 

9.954823 
.954762 

1.02 

9.682063 

.682387 

5.40 

500 

10.317937 
.317613 

19 

18 

43 

44 
45 

.637411 
!  687678 

.00 
4.37  ' 
4.37 

.954701 
.954640 
.9.54579 

1.02 
1.02 

.682710 
.683033 
.683356 

.00 
5.38 
5.38 

.317290 
•  .316967 
.316644 

17 
16 
15 

46 

!  638197 

4.37 

4  OK 

.954518   J'iS 

.683679 

5.38 

K.  Q'? 

.316321 

14 

47 

.638458   *-*i 

.954457 

1  .US 

.684001 

a.6f 
500 

.315999 

13 

48 

.638720   T'S 

.954396   i'^S 

.684324 

.00 

.315676 

12 

49 
50 

.638981 
.639242 

4.OO 

4.35 
4.35 

.954.335 
.954274 

1.02 
1.02 

.684646 
.684968 

5.37 
5.37 
5.37 

.315354 
.315032 

11 
10 

51 
52 

9.639503 
.639764 

4.35 

4QO 

9.954213 
.954152 

1.02 

Ino 

9.685290 
i  .685612 

5.37 

10.314710 

.314388 

9 

8 

53 

.640024 

.OO 

A  fin 

.984090 

.0.3 

.685934   R-'OK 

.314066 

7 

54 

.640284   *•*' 

.954029 

1.02 

COBCIK"       O.OO 

.Ohb2oo     t  nr, 

.313745 

6 

55 

.640544  i  7'SJ 

.953968   i'Xo 

.686577 

U.OI 

.313423 

5 

56 

.640804 

t.Oit 

.953906   i'X2 

.686898 

.313102 

4 

57 

.641064 

4.33 

.953845 

J  .VX 

.687219 

5.35 

.312781 

3 

58 
59 

.641324 
.641583 

4.  '32 

4QO 

.953783 
.953722 

1  .03 
1.02 

*  no 

.687'540 

.687861 

5.35 
5.35 

**  QK 

.312460 
.312139 

2 
1 

60 

3.  641842 

.OA 

9.953660   llVO 

9.688182 

10.311818 

0 

' 

Cosine. 

D.  r.  j 

Sine.   D.  1". 

Cotang. 

D.  1".    Tang.    ' 

28° 


COSINES,  TANGENTS,   AND  COTANGENTS. 


153< 


' 

j  Sine. 

D.  1". 

Cosine. 

D.r. 

Tang. 

D.r. 

Cotang. 

i 

0 

i  9.641842 

9.953660 

i  An 

9.688182 

500 

10.311818 

60 

1 

.042101 

4.32 

4Q>> 

.953599   J'XS 

.688502 

.00 
K  00 

.311498 

59 

.642360 

.O* 

4QA 

.953537 

J.  .  Vt> 

1"  fiQ 

.688823 

•>.'<„  . 

500 

.311177 

58 

Jj 

.642618 

.OU 

.953475 

.Uo 
Ino 

.6891-13 

.00 

500 

.310857 

57 

4 

.042877 

4.32 

4QA 

.95:3413 

.Uo 

.689463 

.00 
500 

.310,537 

56 

.043135 

.oU 
4  on 

.953352 

Ino 

.689783 

.00 

500 

.310217 

55 

6 

.043393 

.oU 

.953290 

.Uo 

1  03 

i  .690103 

.00 

K   OO 

.309897 

54 

7 

>  .043650 

A  OA 

.953228 

1  03 

.090423 

O.OO 
5QO 

.309577 

53 

8 

i  .643908   *'£ 

.953166 

.090742 

.04 
500 

.309258 

52 

9 
10 

!  .644165 
.644423 

t.TOQ 

4.30 
4.28 

.953104 

.953042 

1^03 
1.03 

j  .691062 
.691381 

.OO 

5.32 
5.32 

.308938 
.308619 

51 
50 

11 
12 

!  9.644680  i  4  „. 
i  .644936   I'S 

9.952980 
.952918 

1.03 

Ins 

!  9.691700 
j  .692019 

5.32 

r  •>«> 

10.308300 
.307981 

49 

48 

13 
14 

.645193 
.615450 

t.xo 
4.28 

.C52855 
.952793 

.UO 

1.03 

.6923.8 
.692656 

5.  Ox! 
5.30 

5QO 

.307662 
.307344 

47 
46 

15 
16 

17 
18 

.015706 
.645962 
.646218 
.046474 

4.27 
4.27 
4.27 

4.27 

A  OK 

.952731 
.952669 
.952000 
.952544 

1.'03 
1.05 
1.03 

Ine 

.692975 
.693293 
.693612 
.693930 

.04 

5.30 
5.32 
5.30 

.307025 
.306707 
.300388 
.300070 

45 
44 
43 
42 

19 

.640729   ?•££ 

.952481 

.UO 
Ino 

.694248 

5.30 
Son 

.305752 

41 

20 

!  .046984   *:27 

.952419 

.Uo 

1.05 

.694500 

.OU 

5.28. 

.305434 

40 

21 
22 
23 
24 

25 
'  26 

27 
28 
29 
30 

9.047240 
.647494 
!  .647749 
'  .648004 
.048258 
.048512 
.648760 
i  .049020 
.049274 
.649527 

4.23 
4.25 
4.25 
4.23 
4.23 
4.23 
4.23 
4.23 
4.22 
4.23 

9.952356 
.952294 
.952231 
.952168 
.952106 
.952043 
.951980 
.951917 
.951854 
.951791 

1.03 
1.05 
1.05 
1.03 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 

1  9.094883 
.695201 
.095518 
.695836 
.696153 
.696470 
,696787 
.697103 
.697420 
.697736 

5.30 
5.28 
5.30 
5.28 
5.28 
5.28 
5.27 
5.28 
5.27 
5.28 

10.305117 
.304799 
.304482 
.304164 
.303847 
.303530 
.303213 
.302897 
.302580 
.302264 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.649781 
.650034 
.650287 
.650589 
.650792 
.651044 
.651297 
.651549 
.651800 
.652052 

4.22 
4.22 
4.20 
4.22 
4.20 
4.22 
4.20 
4.18 
4.20 
4.20 

9.951728 
.951665 
.951602 
.951539 
.951470 
.951412 
.951349 
.951280 
.951222 
.951159 

1.05 
1.05 
1.05 
1.05 
1.07 
1.05 
1.05 
1.07 
1  05 

iLoa 

9.698053 
.698369 
.698685 
.699001 
.69931(5 
.699032 
.699947 
.700263 
.700578 
.700893 

5.27 
5.27 
5.27 
5.25 
5.27 
5.25 
5.27 
5.25 
5.25 
5.25 

10.301947 
.301631 
.301315 
.300999 
.300684 
.300368 
.300053 
.299737 
.299422 
.299107 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 

9.652304 
.652555 
.652800 
.053057 
053308 
.053558 

4.18 
4.18 
4.18 
4.18 
4.17 

9.951090 
.951032 
.850968 

.950905 
.950841 
.950778 

1.07 

1.07 
1.05 
1.07 
1.05 
107 

9.701208 
.701523 
.701837 
.702152 

.702406 
.702781 

5.25 
5.23 
5.25 
5.23 
5.25 

f  OO 

10.298792 
.298477 
.298163 
.297848 
.297534 
.297219 

19 
18 
17 
16 
15 
14 

47 
48 
49 
50 

.053808 
.654059 
.05430!) 
.654558 

4.17 
4.18 
4.17 
4.15 
4.17 

.950714 
.9.50050 
.958586 
.950522 

.VI 

1.07 
1.07 
1.07 
1.07 

.703095 
.703409 
.703722 
.704036 

O.xJo 

5.23. 
5.22 
5.23 
5.23 

.296905 
.296591 
.296278 
.295964 

13 
12 
11 
10 

51 

9.654808 

44|M 

9.950458 

9  704350 

10.295650   9 

52 
63 

54 
55 

56 
57 
58 
59 
60 

.655058 
.655307 
.655556 
.655805 
.650054 
056302 
.656551 
.656799 
9.657047 

.17 
4.15 
4.15 
4.15 
4.15 
4.13 
4.15 
4.13 
4.13 

.950394 
.950,330 
.950200 
.950202 
.950138 
.950074 
.950010 
.949945  I 
9.949881 

l!()7 
1.07 
1.07 
1.07 
1.07 
1.07 
1.08 
1.07 

!  704663 
.704976 
.705290 
.705603 
.705916 
.706228 
.706541 
.706854 
9.707166 

5.22 
5.22 
5.23 
5.22 
5.22 
5.20 
5.22 
5.22 
5.20 

.295337  ! 
.295024  ! 
.294710 
.294397 
.294084 
.293772 
.293459 
.293146 
10.292834 

8 
7 
6 
5 
4 
3 
2 
1 
0 

'  \  Cosine.   D.  1". 

Sine,  ;  D.  1". 

Cotang. 

D.  1". 

Tang. 

i 

385 


63C 


27< 


TABLE  XXV.— LOGARITHMIC  SINES, 


152° 


i 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 

9.657047 
.657295 
.657542 

.13 
.12 

1Q 

9.949881 
.94!  (81  6 
.949752 

1.08 
1.07 

9.707166 

.70747'8 
.707790 

5.20 
5.20 

10.292&34 
.292522 

.292210 

60 
59 
58 

3 

4 
5 
6 
7 
8 

.657790 
.658037 
.658284 
.658531 
.658778 
.659025 

'  .14 
.12 
.12 
.12 
.12 
.12 
i  A 

.949688 
.949623 
.949558 
.949494 
.949429 
.949364 

1  .07 
1.08 
1.08 
1.07 
1.08 
1.08 

.708102 
.708414 
.708726 
.709037 
.709349 
.709660 

5.20 
5.20 
5.20 
5.18 
5.20 
5.18 
5  -to 

.291898 
.291586 
.891274 
.290963 
.290651  . 
.290340 

56 
55 
54 
53 
52 

9 

.659271 

1  .lu 

.949300 

1  .07 

.709971 

.1(5 

.29002!) 

51 

10 

.  659517 

1  .10 
.10 

.949235 

1.08 
1.08 

.710282 

5.18 
5.18 

.289718 

50 

11 
12 

9.659763 
.660009 

.10 

[  9.949170 
!  .949105 

1.08 

9.710593 
.710904 

5.18 

10.289407 

.28901)6 

49 

48 

13 

.660255 

1  .  10 

1ft 

.949040 

1.08 

1AQ 

.711215 

5.18 

r  i  " 

.288785 

47 

14 

.660501 

.lu 

AQ 

.948975 

.Uo 

.711525 

0.17 

.288475 

46 

15 

.660746 

<  .08 

Ai2 

.948910 

1.08 

1AQ 

.711&36 

5.18 

.288164 

45 

16 
17 

18 
19 

.660991 
.661236 
.661481 
.661726 

<  .03 

4.08 
4.08 
4.08 

4(Y? 

i-  .948845 
i  .948780 
!  .948715 
1  .948650 

.08 

1.08 
1.08 
1.08 

.712146 
.712456 
.712766 
.713076 

5.17 
5.17 
5.17 
5.17 

.287854 
.287544 
.  287234 
.286924 

44 
43 

42 
41 

20 

.661970 

.\J( 

4.07 

.948584 

1  .10 
1.08 

.713386 

5.17 
5.17 

.286614 

40 

21 

9.662214 

A  AQ 

9.948519 

1AQ 

9.713696 

K  1  X 

10.286304 

39 

22 
23 
24 
25 
26 
27 
28 

.662459 
.662703 
.662946 
.663190 
.6634:33 
.663677 
.663920 

4  .  Uo 
4.07 
4.05 
4.07 
4.05 
4.07 
4.05 

4  A" 

.948454 
.948388 
.948323 
.948257 
.948192 
.948126 
.948060 

.Uo 
1.10 
1.08 
1.10 
1.08 
1.10 
1.10 

.714005 
.714314 
.714624 
.714933 
.715242 
.715551 
.715860 

0.  lO 

5,15 

5.17 
5.15 
5.15 
5.15 
5.15 

.285995 
.285686 
.285376 
.285007 
.2847'58 
.284449 
.884140 

Ajy 

86 

35 
34- 
33 
32 

29 
30 

.664163 
.664406 

.Oo 
4.05 
4.03 

.947995 
.947929 

1  .08 
1.10 
1.10 

.716168 
.716477 

5.13 
5.15 
5.13 

.283832 
.283523 

31 
30 

31 
32 
33 

9.664648 
.664891 
.665133 

4.05 
4.03 

9.947863 
.947797 
.947731 

1.10 
1.10 

9.716785 

.717093 
.717401 

5.13 

5.13 

10.283215 

.282907 
.282599 

29 

28 
27 

34 
35 
36 
37 

.665375 
.665617 
.665859 
.666100 

4.03 
4.03 
4.03 
4.02 

4  no 

.947665 
.947600 
.947533 
.947467 

1.10 
1.08 
1.12 
1.10 

11  A 

.717709 
.718017 
.718325 
.718633 

5.13 
5.13 
5.13 
5.13 

.282291 
.281983 
.281677, 
.281367 

26 
25 
24 
23 

38 
39 
40 

.666342 
.666583 
\666824 

.Uo 
4.02 
4.02 
4.02 

.947401 
.947335 
.947269 

.  1U 
1.10 
1.10 
1.10 

.718940 
.719248 
.719555 

5.12 
5.13 
5.12 
5.12 

.281060 
.280752 
.280445 

1? 

20 

41 
42 
43 
44 
45 
46 
47 
48 
49 

9.667065 
.667305 
.667546 
.667786 
.668027 
.668267 
.668506 
.668746 
.668986 

4.00 
4.02 
4.00 
4.02 
4.00 
3.98 
4.00 
4.00 

9.947203 
.947136 
.947070 
.947004 
.946937 
.946871 
.946804 
.946738 
.946671 

1.12 
1.10 
1.10 
1.12 
1.10 
1.12 
1.10 
1.12 

9.719862 
.720169 
.72047'6 
.720783 
.721089 
.7'21396 
.721702 
.722009 
.762315 

5.12 
5.12 
5.12 
5.10 
5.12 
5.10 
5.12 
5.10 

10.280138 
.279831 
.279524 
.279217 
.278911 
.278604 
.278298 
.277991 
.277685 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

.669225 

3.98 
3.98 

.946604 

1.12 
1.10 

.722621 

5.10 

5.10 

.277379 

10 

51 
52 

9.669464 
.669703 

3.98 

9.946538 
.946471 

1.12 

9.722927 

.723232 

5.08 

10.277073 

.276768 

9 
8 

53 

.669942 

3.98 

3f\Q 

.946404 

1.12 

11O 

.728538 

5.10 

51  A 

.276462 

7 

54 

55 
56 
57 
58 
59 
CO 

.670181 
.670419 
.670658 
.670896 
.671134 
.671372 
9.671609 

.\K) 

3.97 
3.98 
3.97 
3.97 
3.97 
3.95 

.946337 
.946270 
.946203 
.946136 
.946069 
.946002 
9.945935 

.1* 
1.12 
1.12 
1.12 
1.12 
1.12  j 
1.12 

.7881844 

.724149 
.724454 
.7'24760 
.725065 
.725370 
'  9.725674 

.1U 

5.08 
5.08 
5.10 
5.08 
5.08 
5.07 

.276156 
.275351 
.273546 

.275240 
.274935 
.274630 
10.274326 

G 
5 
4 
3 

2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r.  1 

Cotang. 

D.  1". 

Tang. 

1 

117C 


3SG 


COSINES,   TANGENTS,  AND  COTANGENTS. 


151' 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 

3 
4 
5 
6 

8 
9 
10 

9.671609 
.671847 
.672084 
.672321 
.672558 
.672795 
.673032 
.673268 
.673505 
.673741 
.673977 

3.97 
3.95 
3.95 
3.95 
3.95 
3.95 
3.93 
3.95 
3.93 
3.93 
3.93 

9.9459,35 
.945868 
.945800 
.945733 
.945666 
.945598 
.945531 
.945464 
.945396 
.9453S8 
.945261 

1.12 
1.13 
1.12 
1.12 
1.13 
1.12 
1.12 
1.13 
1.13 
1.12 
1.13 

9.725674 
.735979 

.726284 
.726588 
.726892 
.727197 
.7'27501 
.7'27805 
.7-28109 
.728412 
.7'28716 

5.08 
5.08 
5.07 
5.07 
5.05 
5.07 
5.07 
5.07 
5.05 
5.07 
5.07 

10.274326 
.274021 
.273716 
.273412 
.273108 
.272803 
.272499 
.272195 
.271891 
.271588 
.271284 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 

9.674213 
.67-4448 
.674684 
.674919 

3.92 

3.93 
3.92 

9.945193 
.945125 

.945058 
.944990 

1.13 
.12 
.13 

9.729020 
.729323 
.729626 
.729929 

5.05 

5.05 
5.05 

10.270980 
.270677 
.270374 
.27-0071 

49 
48 
47 
46 

15 
16 
17 

.G7T)  155 
.675390 
.675624 

3.93 
3.92 
3.90 

.944922 
.944854 

.944786 

.18 
.13 
.13 

.730233 

.730535 
.730838 

5.07 
5.03 
5.05 

.269767 
.269465 
.269162 

45 

44 
43 

18 
19 

.675859 
.676094 

3.92 
3.92 

.944718 
.944650 

.13 
1.13 

.731141 
.731444 

5.05 
5.05 

.268859 
.268556 

42 
41 

20 

.676328 

3.90 
3.90 

.944582 

1.13 
1.13 

.731746 

5.03 
5.03 

.268254 

40 

21 
22 
23 
24 
25 
26 
27 

9.676562 

.676796 
.677030 
.677264 
.677498 
.677731 
.677964 

3.90 
3.90 
3.90 
3.90 

3.88 

3.88 

9.944514 
.944446 
.944377 
.944309 
.944241 
.944172 
.944104 

1.13 
1.15 
1.13 
1.13 
1.15 
1.13 

9.732048 
.732351 
.732653 
.732955 

.783257 
.733558 
.733860 

5.05 
5.03 
5.03 
5.03 
5.02 
5.03 

10.267952 
.267649 
.267347 
.267045 
.266743 
.266442 
.266140 

89 
38 
37 
36 
35 
34 
33 

28 
29 
30 

.678197 
.678430 
.678663 

3.88 
3.88 
3.88 
3.87 

.944036 
.943967 
.942899 

1.13 
1.15 
1.13 
1.15 

.734162 
.734463 

.734764 

5  .  03 
5.02 
5.02 
5.03 

.265838 
.265537 
.265236 

82 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 

9.678895 
.679128 
.679360 
.679592 
.679824 
.680056 
.680288 
.680519 

3.88 
3.87  1 
3.87 
3.87 
3.87 
3.87 
3.85 

9.943830 
.943761 
.943693 
.943624 
.943555 
.943486 
.943417 
.943348 

1.15 
1.13 
1.15 
1.15 
1.15 
1.15 
1.15 

9.7.35066 
.7-35367 
.735668 
.735969 
.736269 
.736570 
.736870 
.737171 

5.02 
5.02 
5.02 
5.00 
5.02 
5.00 
5.02 

10.264934 
.264633 
.264332 
.264031 
.263731 
.263430 
.263130 
.262829 

29 
28 
27 
26 
25 
24 
23 
22 

39 
40 

.680750 
.680982 

3  .  85 
3.87 
3.85 

.943279 
.943210 

1.15 
1.15 
1.15 

.737471 
.737771 

5.00 
5.00 
5.00 

.262529 
.262229 

21 

20 

41 
42 
43 
44 

9.681213 
.681443 
.681674 
.681905 

3.  as 

3.85 
3.85 

9.943141 
.94>:072 
.943003 
.942934 

1.15 
1.15 
1.15 

9.738071 
.738371 
.738671 
.7'38971 

5.00 
5.00 
5.00 

10.261929 
.261629 
.261329 
.261029 

19 
18 
17 
16 

45 

.682135 

3.  as 

.942864 

1.17 

.739271 

5.00 

.260729 

15 

46 
47 

48 

.682365 
.682595 
.682825 

3.83 
3.83 
3.83 

.942795 
.942726 
.942656 

1.15 
1.15 
1.17 

.739570 
.739870 
.740169 

4.98 
5.00 
4.98 

A  no  ^ 

.260430 
.260130 
.259831 

14 
13 
12 

49 

.683055 

3.  as 

.942587 

1.15 

.740468 

4..  98 

.2E9532 

11 

50 

.683284 

3'  83 

.942517 

1.17 
1.15 

.740767 

4!98 

.259233 

10 

51 

9.683514 

9.942448 

9.741066 

4  no 

10.258934 

9 

52 

.683743 

?•§** 

.942378 

1  .17 

.741365 

.Do 

.258635 

8 

58 

54 

.683972 
.684201 

3!  82 

.942308 
.942239 

1  .17 
1.15 

.741664 
.741962 

4  .98 
4.97 

4  no 

.258336 
.258038 

7 
6 

55 

.684430 

3  on 

.942169 

1  .  17 

.742261 

.WO 

.257739 

5 

56 
57 
58 
59 
60 

.684658 
.684887 
.686115 
.685343 
9.685571 

.oO 
3.82 
3.80 
3.80 
3.80 

.942099 
.942029 
.941959 
.941889 
9.941819 

1.17   ; 

1.17  ! 
1.17  ! 
1.17  i 
1.17 

.742559 
.742858 
.743156 
.743454 
9.743752 

4.97 
4.98 
4.97 
4.97 
4.97 

.257441 
.257142 
.256844 
.256546 
10.256248 

4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

i 

TABLE  XXV.— LOGARITHMIC  SINES, 


150° 


' 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

9.685571 
.685799 

3.80 

9.941819 
.941749 

1.17 

9.743752 

.744050 

.97 

10.256248 
.266950 

60] 
59, 

2 
3 
4 
5 

.686027 
.080254 
.680482 
.686701) 

3.78 
3.80 
3.78 

.941679 
.941609 
.941539 
.941469 

1.17 
1.17 

1.17 
1  18 

.744348 
.744645 
.744943 
.745240 

.95 
.97 
.95 

.255652 

'.  255057 
.254760 

58' 
57 
56 
55 

6 
7 
8 
9 
10 

.686936 
.687163 
.687389 
.687616 
.687843 

3.78 
3.77 
3.78 
3.78  , 
3.77  i 

.941398 
.941328 
.941258 
.941187 
.941117 

1.17 
1.17 
1.18 
1.17 
1.18 

.745538 
.7^335 
.74ul32 
.740429 
.746726 

.95 
4.95 
4.95 
4.95 
4.95 

.254462 
.254165 
.253868 
.253571 
.25:32;4 

54 
53 
52 
51 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
23 

9.  638069 
.688295 
.688521 
.688747 
.688972 
.689198 
.689423 
.689648 
.68937'3 
.690098 

3.77 
3.77 
3.77 
3.75 
3.77 
3.75 
3.75 
3.75 
3.75 
3.75 

9.941046 
.940975 
.940905 
.940834 
.940763 
.940693 
.940622 
.940551 
.940480 
.940409 

1.18 
1.17 
1.18 
1.18 
1.17 
1.18 
1.18 
1.18 
1.18 
1.18 

9.747023 
.747319 
.747616 
.747913 
.748209 
.748505 
.748801 
.749097 
.749393 
!  .749689 

4.93 
4.95 
4.95 
4.93 
4.93 
4.93 
4.93 
4.93 
^4.93 
4.93 

10.252977 
.252081 
.252384 
.252087 
.251791 
.251495 
.251199 
.250903 
.260607 
.250311 

49 

48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
25 
27 
28 
29 
30 

9.690323 
.090548 
.090772 
.690996 
.691220 
.691444 
.091668 
.691892 
.692115 
.692339 

,  3.75 
3.73 
3.73 
3.73 
3.73 
3.73 
3.73 
3.72 
3.73 
3.72 

9.940338 
.940267 
.940196 
.940125 
.940054 
.C33982 
.939911 
.939840 
.939768 
.939697 

1.18 
1.18 
1.18 
1.18 
1.20 
1.18 
1.18 
1.20 
1.18 
1.20 

9.749985 
.7'50281 
.750576 
.750872 
.751167 
.751462 
.751757 
.752052 
.752347 
.752642 

4.93 
4.92 
4.93 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 

10.250015 
.249719 
.249424 
.249128 
.248833 
.248538 
.248243 
.247948 
.247053 
.247358 

39 
38 
37 
36 
35 
84 
33 
32 
31 
30 

31 
32 
33 

9.692562 

.692785 
.693008 

3.72 
3.72 

9.939625 
.939554 
.939482 

1.18 
1.20 

9.752937 
.753231 
.753526 

4.90 
4  92 
A  on 

10.247063 
.240709 
.24647'4 

29 

28 
27 

34 
,35 

.693231 
.693453 

3.70 

.939410 
.939339 

1.18 

.75:3820 
.754115 

4.92 

.246180 

.245885 

26 
25 

36 
37 

38 
39 
40 

.693676 
.693898 
.694120 
.694342 

.6945(54 

3.70 
3.70 
3.70 
3.70 
3.70 

.939267 
.939195 
.939123 
.939052 

.938980 

1.20 
1.20 
1.18 
1.20 
1.20 

.751409 
.754703 
.754997 
.755291 

.755585 

4.90 
4.90 
4.90 
4.90 
4.88 

.245591 
.245297 
.245003 
.244709 
.244415 

24 
23 

22 

21 
20 

41 
42 
43 
44 

9.694786 
.695007 
.695229 

.695450 

3.68 
3.70 
3.68 

9.938908 
.938836 
.938703 
.938091 

1.20 
1.22 
1.20 

9.755878 
.756172 
.756465 
.7507'59 

4.90 
4.88 
4.90 

10.244122 
.24=3828 
.243535 
.243241 

19 
18 
17 
16 

45 

46 

.695671 
.695892 

3.68 

.9:38619 
.938547 

1.20 

.757052 
.757345 

4.88 

.242948 
.242655 

15 
14 

47 

.696113 

.938475 

.757038 

A  00 

.242362 

13 

48 
49 

.696834 

.696554 

3.67 

.938402 
.9-38330 

1.20 

!  758224 

4.88 

400 

.242069 
.241776 

12 
11 

50 

.696775 

3.67 

.938258 

t'M 

.758517 

4.88 

.241483 

10 

51 

9.696995 

9.9.38185 

9.758810 

10.241190 

9 

52 

.697215 

.938113 

.759102 

A  QQ 

.240898 

8 

53 
54 
55 

56 
57 
58 
59 

.697435 
.097654 
.097874 
.698094 
.698313 
.698532 
.698751 

3.65 
3.67 
3.67 
3.65 
3.65 
3.65 

.938040 
.937907 
.937895 
.937822 
.937749 
.937070 
.937604 

1.22 
1.20 
1.22 
1.22 
1.22 
1.20 

.75931)5 
.759087 
.759979 
.700272 
.700504 
.760856 
.701148 

4.87 
4.87 
4.88 
4.87 
4.87 
4.87 

.240605 
.240313 
.240021 
.239728 
.239436 
.239144 
.238852 

7 
6 
5 
4 
3 
2 
1  • 

60 

9.698970 

9.937531 

1.22 

9.761439 

4.85  j 

10.238561 

0 

'  I 

Cosine. 

D.  1". 

Sine. 

D.  r.  : 

Cotang. 

D.  r.  i 

Tang. 

' 

119= 


QQQ 


COSINES,  TANGENTS,   AND  COTANGENTS. 


149° 


/ 

Sine. 

D.r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

/ 

0 

9.G98970 

3/»~ 

9.937531 

OO 

9.761439 

4OQ 

10.238561 

CO 

1 

.099189 

.UO 
3ftft 

.937458 

JG3 

oo 

.761731 

.no 
4'    W7 

.238209 

59 

2 

.699407 

.Oo     ; 

.937385 

•jEo 

.762023 

.o< 

.237977 

58 

3 

.699626 

3.05 

q  eq 

.937312 

.22     i 

Oq 

.762314 

4.85 

A     QW 

.237686 

57 

4 

.699844 

o.  Do 

Q    |;O 

.937238 

.^O 
OO 

.762606 

4.  o< 

A      OK 

.237394 

56 

5 

.700062 

o.Uo 

3C»o       ; 

.937165 

ijPe 

OO 

.762897 

4.OO 

4Qff 

.237103 

55 

G 

.700280 

.  UO         i 
o    rf»Q         1 

.937092 

.3960 

OO 

.763188 

.00 

.236812 

64 

7 
8 
I 

.700498 
.700716 
.700933 

6  .  UO        I 
3.03 

3.02 

o     /»q 

.937019 
.936946 
.936872 

.,£& 
.22 
.%* 

OO 

.763479 
.763770 
.764061 

4.85 
4.85 
4.85 

4  fix 

.236521 
.236230 
.235939 

53 
52 
51 

10 

.701151 

o.  Do 

3.62 

.936799 

JQ0 
.23 

.764352 

.00 

4.85 

.235648 

50 

11 

9.701368 

o   /?o 

9.936725 

oo 

9.764643 

400 

10.235357 

49 

12 

.7  1585 

o.O/i 

.936652 

.  .  - 
OQ 

.764933 

.OO 
4QK 

.235067 

48 

13 
14 
15 

.701802 
.702019 
.702236 

3.02 
3.62 
3.62 

0    ti.  A 

.93(5578 
.936505 
.936431 

.^o 

.22 
.23 
oq 

.765224 
.765514 
.765805 

.OO 

4.83 
4.85 

400 

.234776 
.234486 
.234195 

47 
46 
45 

16 

.702452 

o.  OU 

O     |*O 

.936357 

.  /*o 
fc» 

.766095 

.  OO 
4OQ 

.233905 

44 

17 

.702669 

Io  .  u* 
0     «A 

.936284 

./££ 
oq 

.766385 

OO 
400 

.233615 

43 

18 

.702885 

o.OU 
q   I*A 

.936210 

.*O 

oq 

.766675 

.OO 

4QO 

233325 

42 

19 

.703101 

o.OU 

0     (»A 

.936136 

.^o 
oq 

.766965 

.OO 
4QQ 

1233035 

41 

20 

.703317 

fj  .  OU 

3.60 

.936062 

.  4>fj 

.23 

.767255 

.00 

4.83 

.232745 

40 

21 

9.703533 

3fiO 

9.9a5988 

•     OQ 

9.767545 

4QO 

10.232455 

39 

22 

.703749 

.  DU 

3     CO 

.935914 

.-^O 

oq 

.767834 

.0x5 
4QQ 

.232166 

38 

23 

.703964 

.Do 

3CQ 

.935840 

.  --•> 
OQ 

.768124 

.OO 
4QQ 

.231876 

37 

24 

.704179 

.  UO 
3(*C\ 

.935766 

.  &j 
OQ 

.768414 

.OO 

.231586 

36 

25 

.704395 

.00 

3EQ 

.935692 

.    ./CO 

oq 

.768703 

4.82 

4QO 

.231297 

35 

26 

.701610 

.Oo 

q     tO 

.935618 

.  ,'8RJ 

OfC 

.768992 

.ox5 

4QO 

.231008 

34 

27 

.704825 

O.OO 
0     KQ 

.935543 

,^o 
oq 

.769281 

.t52 
4QQ 

.230719 

33 

28 

.705040 

o  .Do 
3x.>y 

.935469 

,ao 

OQ 

.769571 

.OO 

4vO 

.230429 

32 

29 

.7052,54 

.  Of 
q  KQ 

.935395 

.*<^O 
OK 

.769860 

.0/«& 
4ttrt 

.230140 

31 

30 

.705469 

O.  Do 

3.57 

.935320 

.-^0 

.23 

.770148 

.ol/ 

4.82 

.229852 

30 

31 

9.705683 

3F=Q 

9.9-35246 

on 

9.770437 

4QO 

10.229563 

29 

89 

.705898 

.  Do 

3    erf 

.983171 

.  vO 
OQ 

.770726 

.Osi 
4QO 

.229274 

28 

33 

.700112 

.y  t 

3SfV 

.  935097 

.itRS 

OS        1 

.771015 

.OX 

.228985 

27 

34 
35 

.706326 
.706539 

.5* 
3.55 

3     try 

.9.35022 
.934948 

.xiO 

.23     i 

OK 

.771303 
.771592 

4.80 
4.82 

4QA 

.228697 
.228408 

26 
25 

36 

.706753 

.  Ol 
q    E7 

.934873 

.^a 

OK 

.771880 

.oO 

4Qf\ 

.228120 

24 

37 
38 

.706967 
.707180 

O  .  Ol 

3.55 

S.cr: 

.934798 
.934723 

./^O 

.25 

00 

.772168 
.772457 

.OU 

4.82 

4OTV 

.227832 
.221543 

23 
22 

39 

40 

.707393 
.707606 

OO 

3.55 
3.55 

.934649 
.934574 

.-^0       : 

.25 
.25 

.772745 

.773033 

.  OU 

4.80 
4.80 

.227255 
.226967 

21 
20 

41 
42 

9.707819 
.708032 

'3.55 

3e~ 

9.934499 
.934424 

.25 

9.773321 

.773608* 

4.78 

10.22667'9 
.226392 

19 

18 

43 

.708245 

.OO 
3K.K 

.934349 

.25 
ok 

.773896 

4.80 

4QA 

.226104 

17 

44 

.708458 

,OO 

3cq 

.934274 

./&> 

OFC 

.774184 

.OU 

4r*o 

.225816 

16 

45 

.708670 

.00 

3eq 

.934199 

.!&) 
O^ 

.774471 

.  *O 

4QA 

.225529 

15 

46 
47 

48 

.708882 
.709094 
.709306 

.00 

3.53 
3.53 

3    tea 

.934123 
.934048 
.933973 

.  £i 
.25 
1.25 

1OK 

.774759 
.775046 
.775333 

.oU 
4.78 
4.78    • 

40A 

.225241 
.224954 

.224667 

14 
13 
12 

49 
50 

.709518 
.7097:30 

.  OO 

3.53 
3.52 

.933898 
.933822 

.to 
1.27 
1.25 

.775621 

.775908 

.oU 

4.78 
4.78 

.224379 
.224092 

11 

10 

51 
52 
53 
54 
55 
56 
57 
58 
'  59 
CO 

9.709941 
.710153 
.710364 
.710575 

.710786 
.710997 
.711208 
.711419 
.711629 
9.711839 

3.53 
3.52 
3.52 
3.52 
3.52 
3.52 
3.52 
3.50 
3.50 

9.933747 
.933671 
.9:33596 
.933520 
933445 
.933369 
.933293 
.933217 
.933141 
9.932066 

1.27 
1.25 
1.27 
1.25 
1.27 
1.27 
1.27 
1.27 
1.25 

9.776195 
.776482 
.776768 
.777055 
.777'342 
.777628 
.777915 
.778201 
.778488 
9.778774 

4.78 

4.77 
4.78 
4.78 
4.77 
4.78 
4.77 
4.78 
4.77 

10.223805 
.223518 
.223232 
.222945 
.222658 
.222372 
.222085 
.221799 
.221512 
10.221226 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

/ 

Cosine. 

D.  I',    i 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

/ 

389 


SI- 


TABLE  XXV.-LOGARITHMIC  SINES, 


K8' 


I 

' 

Sine. 

D.  1". 

|   Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.711839 

9.933066 

i 

9.778774 

10.221226 

60 

1 

2 
3 
4 
5 

.712050 
.712260 
.712469 
.712679 

.712889 

3^50 
3.48 
3.50 
3.50 

3AQ 

.932990 
.932914 
.932838 
.932762 
.932685 

I  '.27 
1.27    ! 
1.27     ! 

1.28 

.779060 
.77984(5 
.779632 
.779918 
.780203 

4.77 

4.77 
4.77 
4.77 
4.75 

.220940 
.220654 
.220368 
.220082 
.219797 

59 
58 
57 
56 
55 

6 

7 

.713098 
.713308 

,4o 

3.50 

340 

.932609 
.932533 

1  i27* 

.780489 
.780775 

4.77 

4.77 

.219511 
.219225 

54 
53 

8 
9 
10 

.713517 
.713726 
.713935 

.4o 

3.48 
3.48 
3.48 

.932457 
.932380 
.932304 

i!sj 

1.27 
1.27 

.781060 
.781346 
.781631 

4  .75 
4.77 
4.75 
4.75 

.218940 
.218654 
.218369 

52 
51 
50 

11 
12 
13 
14 

9.714144 
.714352 
.714561 
.714769 

3.47 
3.48 
3.47 

o   40 

9.932228 
.932151 
.932075 
.931998 

1.28 
1.27 
1.28 

9.7'81916 

.782201 
.782486 
.7^2771 

4.75 
4.75 
4.75 

10.218084 
.217799 
.217514 
.217229 

49 

48 
47 
46 

15 

16 

.714978 
.715186 

o  .4o 

3.47 

3/1*7 

.931921 
.931845 

l!*7 

.783C56 
.783341 

4.75 

4.75 

.216944 
.216659 

45 
44 

17 

18 
19 
20 

.715394 
.715602 
.715809 
.716017 

.4* 

3.47 
3.45     1 
3.47 
3.45 

.931768 
.931691 
.931614 
.931537 

i.ls 

1.28 
1.28 
1.28 

.783626 
.783910 
.784105 
.784479 

4.75 
4.73 
^4.75 
4.73 
4.75 

.216374 
.216080 
.215805 
.215521 

43 
42 
41 
40 

21 

9.716224 

347 

9.931460 

1    9ft 

9.784764 

4r-o 

10.215236 

39 

22 
23 

.716432 
.716639 

,4i       ; 

3.45 

3JJC 

.931383 
.931306 

1  .  -vO 

1.28 

.765048 
.785832 

.  to 
4.73 

4i~Q 

.214952 
.214668 

38 
37 

24 

.716846 

.40 

.931229 

1    OQ 

.1  8:61(5 

.  (O 

.214384 

36 

25 

.717'053 

3/1Q       ' 

.931152 

J  .-co 
1    98 

.785900 

4.73 

4r-o 

.214100 

35 

26 

.717259 

.4o 

33*3* 

.931075 

1  .  <^O         j 

.786184 

.  to 

.213816 

34 

27 

.717466 

.40 

o  jrc 

.930998 

1  *28 

.786468 

4.73 

4ft"Q 

.213532 

33 

28 
29 

.717673 

.717879 

3^43 

3/1Q 

.930921 
.930843 

l^SO 

.786752 
.787036 

.  to 

4.73 

4r*o 

.213248 
.212964 

32 
31 

30 

.718085 

,4u 

3.43 

.930766 

l'.30 

.787319 

.  fx 

4.73 

.212681 

SO 

31 

9.718291 

3,40 

9.930688 

9.787603 

10.212397 

29 

32 

as 

.718497 
.718703 

.4o 

3.43 

Q     A*) 

.930611 
.930533 

l!30 

.  7  87886 
.7*8170 

4.72 
4.73 

.212114 
.211830 

28 
27 

34 

.718909 

o  .  4o      ! 

.930456 

1  -~°      i 

.788453 

4.72 

.211547 

26 

35 

.719114 

3     An        \ 

.930378 

*•«•*• 

.788736 

4.72 

.211264 

25 

36 

.719320 

.4o 
349 

.930300 

1   28 

.789019 

j'wn 

.210981 

24 

37 

.719525 

.4,« 
3     An 

.930223 

'on 

.789202 

4.  t/i 

.210698 

23 

38 
39 

.719730 
.719935 

.4^ 
3.42 

3      1O 

.930145 
.930067 

"30 

on 

.789585 
.789868 

4^72 

.210415 
.210132 

22 

21 

40 

.7'20140 

.4/5 

3.42 

.929989 

oU 

.30 

.790151 

4.72 
4.72 

.209849 

20 

41 

9.7'20345 

3Af\ 

9.929911 

30 

9.790434 

10.209566 

19 

42 

.7'20549 

.4U 

3A£) 

*  .929833 

.790716 

A     l~-> 

.209284 

18 

43 

44 
45 
46 

.720754 
.720958 
.721163 
.721366 

.4/4 

3.40 
3.40 
3.40 

.929755 
.929677 
.929509 
.929521 

!30 
30 
.30 

.790999 
.791281 
.791563 
.791846 

4.  i£ 

4.70 
4.7'0 
4.72 

.209001 
.208719 
.208437' 
.208154 

17 
16 
15 
14 

47 

.721570 

3.40 

.929442 

'on 

.792128 

4.70 

4r-n 

.207872 

13 

48 

.721774 

3.40 

O    Af\ 

.929364 

.  oU 
on 

.792410 

.  l(J 

.£07590 

12 

49 

.721978 

d.40 

.929286 

.oU 

.792692 

r"\ 

.207308 

11 

50 

.72:2181 

3.38 
3.40 

.929207 

'30 

.792974 

4.  70 

.207026 

10 

51 

9.722385 

3QQ 

9.929129 

32 

9.793256 

4  70 

10.206744 

9 

52 

.722588 

.00 
O    OQ 

.929050 

'on       ! 

.7935:38 

4AQ 

.206462 

8 

53 
54 
55 
56 

57 
58 
59 

.722791 
.7'22994 
.723197 
.723400 
.723603 
.723805 
.724007 

O.OO 

3.38 
3.38 
3.38 
3.38 
3.37 
3.37 

.928972 
.928893 
.928815 
.928736 
.928657 
.928578 
.928499 

.OU 

.32 

iss 

l'32 
1.32 

.71)3819 
.794101 
.794383 
.794664 
.794916 
.795227 
.795508 

.00 
4.70 
4.70 
4.68 
4.70 
4.68 
4.68 

4f»Q 

.206181 
.205899 
.205617 
.205336 
.205054 
.204773 
.204492 

6 
5 
4 
3 
2 
1 

60 

9.724210 

3.38 

9.928420 

9.  79578  J 

.OO 

10.204211 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.I". 

Tang. 

' 

58= 


COSINES,  TANGENTS,   AND  COTANGENTS. 


147= 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0  9.724210   o  o~    9.928420 

i  ofi 

9.795789 

4Afi 

10.204211   GO 

1 
2 

.724412  !  2-Si     .928342 
.724614  i  ggi  !   .928283 

I  .oU 

1.32 

-1   OO 

.79607'0 
!79C851 

.  Oo 

4.68 

4f*  Q 

.203930  j  59 
.203649  58 

3 

.7'2-!8l6  !  o-'o- 

.988183   J-2| 

.796632 

.UO 

.203368  57 

4 
5 

.7-25017 
.725219 

|     O  .  00 

3.37 

3  ME 

.928104   HO 
.928025   J'S 

.796913 
.797194 

4i08 
4  .b< 

.203087  56 
.202806  55 

6 

.725420 

.OO 

30*7 

.9271)46 

J  .tj/f 

.797474 

4AQ 

.202526  54 

8 
9 

.725622 
.7-25823 
.726024 

.01 

:  3.35 
3.35 

O  Of 

.927867 
.927787 
.927708 

li33 
1.32 

.797755 
.798036 
.798316 

.  1x5 
4.68 
4.67 

4A1** 

.202245 
.201964 
.201684 

53 
52. 
51 

10 

.726225   o'^ 

.  92762  J 

1.'33 

.798596 

.  O< 

4.68 

.201404  50 

11   9.720426 

9.927549 

100 

9.798877 

10.201123 

49 

12   .720626 
13  1  .726827 

3^35 

300 

.927470 
.927'390 

»UN9 

i.as 
100 

.799157 
.799437 

4.67 
4.67 

4/»rf 

.200843 
.200563 

48 
47 

14  !  .7'2r027 

.  OO 

3   OK 

.927'310 

.00 

IQO 

.799717 

.  Ol 

.200283  46 

15   .727228 

.00 

300 

.927231 

.••Si 

.799997 

.z 

.200003 

45 

16   .727428 

.00 

.927151 

Ion 

.800277 

4.67 

.199723 

44 

17   .7'27628  j  °-™ 

.927071 

.00 

.800557 

4.67 

.199443 

43 

IS   .727828   2-S 
19   .723027   S-2 

.926991 
.926911 

1  .33 
1.33 

.800836 
.801116 

4.65 

4.67 
4r*t* 

.199164 

.198884 

42 
41 

20 

.728227 

|;5  |   .926831 

l!33 

.801396 

.u< 
4.65 

.198604 

40 

21   fl.  728437 
22   .723026 
23   .728825 

3.32 
3.32 

9.926751 
.926671 
.926591 

1.33 
1.33 

IQO 

9.801675 
.801955 

.802234 

4.67 
4.65 

A  f*~ 

10.198325 
.198045 
.197766 

39 

38 
37 

24   .729024 

0   QO 

.926511 

.OO 

IQO 

.802513  I  ?•!» 

.197487 

36 

25   .729223 

o  .  BBS 

300 

.926431 

.OO 
IQO 

.802792 

4/>r» 

.197208 

35 

26  I  .729422 

.  >>•* 

3  Ml 

.926351 

.OO 
1QK 

.803072 

.u< 

.196928 

34 

27   .  729621 
28   .729820 
29  !  .730018 

.9m 

3.32 
3.30 

3Qo 

.  926270 
!  .926190 
.926110 

.OO 

1.33 

i.a3 

Ifur 

!$08851 

.803630 
.803909 

4^65 
4.65 

.196649 
.196370 
.196091 

33 
32 
31 

30   .730217 

,O« 

3.30 

.926029 

.OO 

1.33 

.804187 

4.63 
4.65 

.195813 

30 

31 

9  730415 

9.925949 

1  ort 

9.804466 

10.195534 

29 

32   .730013 
33  i  .730811 
34  !  .731009 

siao 

3.30 

3AQ 

.925868 
.925788 
.925707 

li'33 
1.35 

1QK 

.804745 
.805023 
.805302 

4.65 
4.63 
4  (55 

.195255 
.194977 
.194698 

28 
27 
26 

35 

.731203 

.40 
3QA 

.925626 

.OO 

.805580 

4.63 
4/»- 

.194420  25 

36 
37 
38 

.731404 
.731602 
.7'31799 

.oU 

3.30 

3.28 

0  OQ 

.925.545 
.925465 
.92538-4 

1^33 
1.35 

IQs: 

.805859 
.803137 
.806415 

.00 
4  63 
4.63 

4/>0 

.194141 
.193863 
.193585 

24 
23 

22 

39  !   7-31996 

O  .  «O 

.925303 

.OO   i 
IQrt 

.806693 

.00 

.19:3307 

21 

40   .732193 

3^28 

.925222 

.OO 

1.35 

.806971 

4.63 
4.63 

.193029 

20 

41   9.732390 

9.925141 

10- 

9.807249 

10.192751 

19 

42   .7-32587   2'Sl 

.925060 

.00 

.807'527 

4.63 

.19247'3 

18 

43 
44 

.7-32784   »•- 
.732980   *•*• 

.924979 
.924897 

1  .35  | 
1-37  i 

10-     t 

.807805 
.80S083 

4.63 
4.63 

.192195 
.191917 

17 

16 

45 
46 

.733177   *-~2 

.7-33:373  I-**, 

.924816 
.9247:35 

.A) 
1.85 

IQX    ! 

.808361 
1808688 

4.63 
4.62 

.191639 
.191362 

15 
14 

47 
4S 
49 
50 

.7*3569 
!  733765 

.733961 
.734157 

O.j*f    : 

3.27 
3.27  i 
3.27 
3.27  ; 

.924654 
.924572 
.924491 
.924409 

.00  ] 
1.37  ! 
1.35 
1.37 
1.35 

.808916 
.809193 
.809471 
.809748 

4.63 
4.62  • 
4.63 
4.62 
4.62 

.191084 
.190807 
.190529 
.190252 

13 

12 
11 

10 

51 
52 
53 

9.734a53 
.734549 
.7347'44 

3  27 

sias 

9.924328 
.924246 
.924164 

1.37 
1.37 

10- 

9.810025 
.810302 
.810580 

4.62 
4.63 

10.189975 
.189698 
.  189420 

9 

8 

54   .734939 

Q  O? 

.924083 

.00 

.810857 

4.62 

.189143 

6 

55   .735135 
56  j  .735:330 
57  i  .735525 
5$   .735719 
59  i  .735914 
60  :  9.7361C9 

3^25 
3.25 
3.23 
3.25 
3.25 

.924001 
.923919 
.923837 
.923755 
.923673 
9.923591 

1  .37 
1.37 
1.37 
1  37 
1.37 
1.37 

.811134 
.811410 
.811687 
.811964 
.812241 
9.812517 

4.62 
4.60 
4.62 
4.62 
4.62 
4.60 

.188866 
.188590 
.188313 
.188036 
.187759 
10.187483 

5 
4 
3 
2 
1 
0 

Cosine.  |  D.  1". 

Sine.   D.  1'.  i  Cotang.  i  D.  1". 

Tang. 

' 

391 


57' 


33° 


TABLE  XXV.— LOGARITHMIC  SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.736109 

300 

9.9-23591 

1Q** 

9.812517 

10.187483 

60 

1 

o 
3 

4 
5 

.736303 
.736498 
.736692 
.736886 
.737080 

."o 

3.25 
3.23 
3.23 

3.23 
300 

.033509 

.923427 
.923345 
.923263 
.923181 

.94 

1.37 
1.37 
1.37 
1.37 

1QO 

.812794 
.81:3070 
.813347 

.813023 
.813899 

4^60 
4.62 
4.60 
4.60 

.1872J6 
.180930 
.  180653 
.180377 
.180101 

59 

57 
56 
55 

6 

7 

.737274 
.737467 

.46 
3.22 

300 

.923098 
.923016 

.OO 

1.37 

1QO 

.814176 
.814452 

4.62 
4.60 

.185824 
.185548 

54 
53 

-8 
9 
10 

.737061 
.737855 
.738048 

JRI 

3.23 
3.22 
3.22 

.922933 
.922851 
.922768 

.OO 

1.37 
1.38 
1.37 

.814728 
.815004 
.815280 

4.60 
4.60 
4.60 
4.58 

.185272 
.184990 
.184720 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 

9.738241 
.738434 
.738627 
.738820 
.739013 
.739206 
.739398 

3.22 
3  22 
3.22 
3  22 
3.22 
3.20 

9.922680 
.922003 
.922520 
.922438 
.923355 
.922272 
.922189 

1.38 
1.38 
1.37 
1.38 
1.38 
1.38 

1QQ 

9.815555 
.815831 

.810107 
.816382 
.816658 
.816933 
.817209 

4.60 
4.60 
4.58 
4.60 
4.58 
4.60 

4MB 

10.184445 
.184169 
.183893 
.183618 
.183343 
.183067 
.182791 

49 
48 
47 
46 
45 
44 
43 

18 

.739590 

300 

.922106 

OO 
10Q 

.817'484 

.OO 
4  fro 

.182516 

42 

19 

.739783 

.££ 

.922023 

.OO 
1QQ 

.817759 

.  *>O 
^  (*ft 

.182241 

41 

20 

.739975 

3^20 

.921940 

.00 
1.38 

.818035 

4.0U 

4.58 

.181965 

40 

21 

9.740107 

390 

9.921857 

1  ''^ 

9.818310 

4  pro 

10.181690 

39 

22 

.740359 

,iiO 

.921774 

1  .Oo 

.818585 

.  Oo 

.181415 

38 

23 
24 

.740550 
.740742 

3.18 
3.20 
3  on 

.921691 
.921607 

1.38 
1.40 

1QQ 

.818860 
.819135 

4.58 
4.58 

4  to 

.181140 

.180865 

37 
30 

25 
26 
27 

.740934 
.741125 
.741316 

•cU 

3.18 
3.18 

.921524 
.921441 
.921357 

.OO 

1.38  ' 
1.40 

1QO 

.819410 
.819684 
.819959 

.00 
4.57 
4.58 

4  pro 

.  180590 
.180316 
.180041 

35 
34 
33 

28 

.741508 

3.~0 

.921274 

.OO 

.820234 

.OO 

.179706 

32 

29 
30 

.741699 
.741889 

3.18 
3.17 
3.18 

.921190 
.921107 

1.40 
1.38 
1.40 

.820508 
.820783 

4.57 
4.58 
4.57 

.179492 
.179217 

31 

30 

31 
32 

9.742080 
.742271 

3.18 

9.921023 
.920939 

1.40 

9.821057 
.821,332 

4.58 

10.178943 
.178668 

29 

28 

33 
34 
35 
36 

.742462 
.742652 
.742842 
.743033 

3  18 
3.17 
3.17 
3.18 

.920856 

.920772 
.920688 
.920604 

1.38 
1.40 
1.40 
1.40 

.821606 
.821880 
.822154 
.822429 

4.57 
4.57 

4.57 
4.58 

.178394 
.178120 
.177840 
.177571 

27 
26 
25 
24 

37 
38 
39 
40 

.743223 
.743413 
.743602 
.743792 

3.17 
3.17 
3.15 
3.17 
3.17 

.920520 
.920436 
.920352 
.920268 

1.40 
1.40 
1.40 
1.40 
1.40 

.822703 

.822977 
.823251 
.823524 

4.57 
4.57 
4.57 
4.55 
4.57 

.177*97 

.177023 

.176749 
.176470 

23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 

9.743982 
.744171 
.744361 
.744550 
.744739 
.744928 
.745117 
.745306 

3.15 
3.17 
3.15 
3.15 
3.15 
3.15 
3.15 
31  o 

9.920184 
.920099 
.920015 
.919931 
.919840 
.919762 
.919677 
.919593 

1.42 
1.40 
1.40 
1.42 
1.40 
1.42 
1.40 

9.823798 
.824072 
.824345 
.824019 
.824893 
.825106 
.825439 
.825713 

4.57 
4.55 
4.57- 
4.57 
4.55 
4.55 
4.57 

4*r?r 

10.176202 
.175928 
.175655 

.175381 
.175107 
.174834 
.174501 
.174287 

19 
18 
17 
16 
15 
14 
13 
12 

49 
50 

.745494 

.745683 

.  lo 
3.15 
3.13 

.919508 
.919424 

l!40 
1.42  i 

.825986 
.826259 

.DO 

4.55 
4.55 

.174014 
.173741 

11 
10 

51 

9.745871 

9.  919339 

9.826532 

10.17-3468 

9 

52 

.746060 

3.15 

31  O 

.919254 

*  49 

.82(5805 

4.55 

.173195 

8 

53 
54 
55 
56 

.746248 
.746436 
.746624 
.746812 

.lo 
3.13 
3.13 
3.13 

39m 

.919169 
.919085 
.919000 
.918915 

l!40 
1.42 
1.42 

.827078 
.827351 
.827'624 
.827897 

4.oo 
4.55 
4.55 
4.55 

4KK 

.172922 
.172649 
.172376 
.172103 

7 
6 
5 
4 

57 

.746999 

.-1* 

.918830 

1  .42 

.828170 

.OO 
A  KQ 

.171830 

3 

58 
59 
60 

.747187 
.747374 
9.747562 

3.13 
3.12 
3.13 

.918745 
.918659 
9.918574 

1  .42 
1.43 
1.42 

! 

.828442 
.828715 
9.828987 

4.5o 
4.55 
4.53 

.171558 
.171285 
10.171013 

2 

1 
0 

'  1 

Cosine. 

D.  1'. 

Sine. 

D.  1".  ! 

Cotang. 

D.  1". 

Tang,  i 

' 

123° 


3C2 


COSINES,   TANGENTS,  AND  COTANGENTS. 


145= 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

!  Tang. 

D.  1". 

Cotang. 

—  i 

0 
1 

9.747562 

.747749 

3.12 
310 

9.918574 
.918489 

1.42 

14O 

9.828987 
.829260 

4.55 
4  5S 

10.171013 
.170740 

60 
59 

2 
3 
4 
5 

.747936 
.748123 
.748310 
.748497 

.Ms 

3.12 
3.12 
3.12 

.918404 
.918318 
.9182:33 
.918147 

.4/6 

1.43 
1.42 
1.43 

.829532 
.829805 
.830077 
.830349 

4.55 
4.53 
4.53 

4KB 

.170468 
.170195 
.169923 
.169651 

58 
57 
56 
55 

6 

.748683 
.748870 

3.10 
3.12 

.918062 
.91797'6 

1  .42 
1.43 

.830621 
.830893 

.OO 

4.53 

4P-Q 

.169379 
.169107 

54 
53 

8 
9 

.749056 
.749243 

3.10 
3.12 

.917891 

.917805 

1  .42 
1.43 

.831165 
.831437 

.OO 

4.53 

4  pro 

.168835 
.168563 

52 
51 

10 

.749429 

3.10 
3.10 

.917719 

!42 

.831709 

:.OO 

4.53 

.168291 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.749615 
.749801 

.749987 
.750172 
.750858 
.750543 

!  750914 
.751099 
.751284 

3.10 
3.10 
3.08 
3.10 
3.08 
3.10 
3.08 
3.08 
3.08 
3.08 

9.917634 
.917548 
.917462 
.91737-6 
.917290 
.917'204 
.917118 
.917'032 
.916946 
.916859 

.43 
.43 
.43 
.43 
.43 
1.43 
1.43 
1.43 
1.45 
1.43 

9.831981 
.832253 
.832525 
.832796 
.833068 
.83.3339 
.833611 
.833882 
.834154 
.834425 

4.53 
4.53 
4.52 
4.53 
4.52 
4.53 
4.52 
4.53 
4.52 
4.52 

10.168019 
.167747 
.167475 
.167204 
.16(5932 
.166661 
.166389 
.166118 
.165846 
.165575 

49 

48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 

9.751469 
.751654 
.751839 
.752023 

.752208 

3.08 
3.08 
3.07 
3.08 

9.916773 
.916687 
.916600 
.916514 
.916427 

1.43 
1.45 
1.43 
1.45 

9.834696 
.834967 
.835238 
.835509 
.835780 

4.52 
4.52 
4.52 
4.52 

10.165304 
.1650:33 
.164762 
.164491 
.164220 

39 

38 
37 
36 
35 

26 
27 

28 
29 
30 

.752392 
.752576 

.752760 
.752044 

.753128 

3.07 
3.07 
3.07 
3.07  ! 
3.07 
3.07 

.916341 
.916254 
.916167 
.916081 
.915994 

1.43 
1.45 
1.45 
1.43 
1.45 
J.45 

.836051 
.836322 
.836593 
.836864 
.837134 

4.52 
4.52 
4.52 
4.52 
4.50 
4.52 

.163949 
.163678 
.163407 
.163136 
.162866 

34 
33 

32 
31 
80 

31 
32 
33 
34 
35 
36 
37 
38 

9.753312 
.753495 
.75:3679 
.753862 
.754046 
.754229 
.754412 
.754595 

3.05 
3.07 
3.07 
3.07 
3.05 
3«05 

3.05 
3f\~ 

9.915907 
.915820 
.915733 
.915646 
.915559 
.915472 
.915385 
.915297 

1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.47 

9.837405 

.&S7675 
.837946 
.838216 
.833487 
.838757 
.839027 
.839297 

4.50 
4.52 
4.50 
4.52 
4.50 
4.  .50 
4.50 

4  ICO 

10.162595 
.162325 
.162054 
.161784 
.161513 
.161243 
.160973 
.160703 

20 
28 
27 
26 
25 
94 
23 
22 

39 
40 

.754778 

.754960 

.Oo 
3.03 
3.05 

.915210 
.915123 

1  .45 
1.45 
1.47 

.839568 
.839838 

.5,6 

4.50 
4.50 

.160432 
.160162 

21 

20 

41 
42 

9.755143 
.755326 

3.05 

9.915035 
.914948 

1.45 

9.840108 
.840878 

4.50 

4-rA 

10.159892 
.159622 

19 
18 

43 

44 

,755508 

.755690 

3.  03 

.914860 
.914773 

1  .47 

1.45 

.840648 
.U0917 

.oU 
4.48 

.159352 
.159083 

17 

16 

45 

.755872 

3.03 

3AQ 

.914685 

1.47 

.841187 

4.50 

.158813 

15 

46 
47 

48 
49 
50 

.756054 
.756236 
.756418 
]  756600 

.756782 

.  Uo 
3.03  i 
3.03 
3.03 
3.03 
3.02 

.914598 
.914510 
.914422 
.914334 
.914246 

1  .45 
1.47 
1.47 
1.47 
1.47 
1.47 

.841457 
.841727 
.841996 
.842266 
.842535 

4^50 
4.48  . 
4.50 
4.48 
4.50 

.158543 
.158273 

.158004 
.  157734 
.157465 

14 
13 
12 
11 
10 

51 

9.756963 

9.914158 

9.842805 

10.1VT195 

9 

52 

.757144 

3.03 

3  no 

.914070 

1.47 

.843074 

4.48 

4AQ 

156926 

8 

53 
54 
55 
56 

57 
58 
59 

.757326 
.757507 

.757688 
.757869 
.758050 
.758230 
.758411 

Uo 

3.02 
3.02 
3.02 
3.02 
3.00 
3.02 
3  An 

.913982 
.913894 
.913806 
.913718  • 
.913630 
913541 
.913453 

1  .47 
1.47 
1.47 
1.47 
1.47 
1.48 
1.47 

.843343 
.843612 
.84:3882 
.844151 
.844420 
.844689 
.844958 

.4o 
4.48 
4.50 
4.48 
4.48 
4.48 
4.48 

.156657 
.156388 
.156118 
.155849. 
.155580 
.155311 
.155042 

6 
5 
4 
3 
2 

1 

60 

9.758591 

.  W 

9.91:3365 

1.47 

9.845227 

4.48 

10.154773 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r.  ! 

Cotang. 

D.  1". 

Tang. 

> 

393 


55° 


35= 


TABLE  XXV.— LOGARITHMIC  SINES, 


i        | 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.758591 

9.913365 

1AQ 

9.845227 

4AQ 

10.154773 

60 

1 

2 

.758772 
.758952 

3.02 
3.00 

.913270 
.913187 

.4o 
1.48 

.845496 
.845764 

:.4O 

4.47 

.154504 
.154236 

59 

58 

3 

.759132 

3.00 

.913099 

1.47 

.846033 

4.48 

.153967 

57' 

4 
5 
6 

.759312 
.759492 
.759672 

3.00 
3.00 
3.00 

.913010 
.912922 
.912833 

1  .48 
1.47 
1.48 

11V 

.846302 
.8-16570 
.846839 

4.48 
4.47 

4.48 

4AQ 

.153698 
.153430 
.153161 

56 
55 
54 

7 
8 
9 

.759852 
.760031 
.760211 

2^98 
3.00 

.912744 
.912655 
.912566 

,4o 
1.48 
1.48 

1AQ 

.847108 
.847376 
.847644 

.4o 
4.47 

4.47 

4AQ 

.152892 
.152624 
.152356 

53 
52 
51 

10 

.760390 

2!  98 

.912477 

.4o 
1.48 

.847913 

.4o 
4.47 

.152087 

50 

11 

9.760569 

9.912388 

9.848181 

4  AW 

10.151819 

49 

12 

.760748 

2(\Q 

.912299 

1AQ 

.848449 

.4i 

.151551 

48 

13 

.760927 

.yo 

O  OQ 

.912210 

.4o 

.848717 

4.47 

4AQ. 

.151283 

47 

14 

.761106 

.912121 

1  .48 

1KA 

.848986 

.4o 

.151014 

46 

15 

.761285 

2.  Jo 

.913031 

.ou 

.849254 

4.47 

.150746 

45 

16 
17 
18 
19 
20 

.761464 
.761642 
.761821 
.761999 
.762177 

2.98 
2.97 
2.98 
2.97 
2.97 
2.98 

.911942 
.911853 
.911763 
.91167'4 
.911584 

1  .48 
1.48 
1.50 
1.48 
1.50 
1.48 

.849522 
.849790 
.850057 
.850325 
.850593 

4.47 
4.47 
4.45 
4.47 
4.47 
4.47 

.150478 
.150210 
.149943 
.149675 
.149407 

44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 

9.762356 
.762534 
.762712 
.762889 
.763067 
.763245 

2.97 

2.97 
2.95 
2.97 
2.97 

9.911495 
.911405 
,911315 
.911226 
.911136 
.911046 

1.50 
1.50 
1.48 
1.50 
1.50 

9.850861 
.851129 
.851396 
.851664 
.851931 
.852199 

4.47 
4.45 
4.47 
4.45 
4.47 

10.149139 

.148871 
.148004 
.148336 
.148069 
.147801 

39 
38 
37 
36 
35 
34 

27 

.763422 

2.95 

2O7 

.910956 

1  .50 

1&A 

.852466 

4.  45 

.147534 

33 

28 

.763600 

.y< 

.910866 

.  .ou 

Irn 

.8527'33 

A  A™ 

.147267 

32 

29 
30 

.763777 

.763954 

2.95 
2.95 
2.95 

.910776 
.910686 

.OU 

1.50 
1.50  . 

.853001 
.853268 

4^45 
4.45 

.146999 
.146732 

31 
30 

31 
32 

9.764131 
.764308 

2.95 

9.910596 
.910506 

1.50 

9.853535 

.853802 

4.45 

10.146465 

.146198 

29 

28 

34 
35 

36 
37 

.764485 
.764662 
.764838 
.765015 
.765191 

2.95 
2.95 
2.93 
2.95 
2.93 

.910415 
.910325 
.910235 
.910144 
.910054 

1  .  52 
1.50 
1.50 
1.52 
1.50 

.854069 
.854336 
.854603 
.854870 
.&55137 

4^45 
4.45 
4.45 
4.4» 

.145931 
.145004 
.145397 
.145130 
.1-14803 

27 
26 
25 
24 
23 

38 

.765367 

2.  Jo 

.909963 

1  .52 
-i  rn 

.855404 

4AZ. 

.144596 

22 

39 

.765544 

2.95 

.909873 

1  ,OU 

.855671 

.  4o 

.144329 

21 

40 

.765720 

2.93 
2.93 

.909782 

1.52 
1.52 

.855938 

4.45 
4.43 

.144062 

20 

41 
42 
43 

9.765896 
.766072 
.766247 

2.93 
2.92 

9.909691 
.909601 
.909510 

1.50 
1.52 

1RO 

9.&56204 
.856471 
.856737 

4.45 
4.43 

4Ae 

10.143796 
.143529 
.143263 

19 
18 
17 

44 
45 
46 

.766423 

.766598 
.766774 

2^92 
2.93 

.909-119 
.909328 
.909237 

.'M 

1.52 
1.52 

.857004 
.857270 
.857537 

.4u 

4.43 
4.45 

.142996 
.142730 
.142463 

16 
15 
14 

47 
48 
49 
50 

.766949 
.767124 
.797800 
.707475 

2.92 
2.92 
2.93 
2.92 
2.90 

.909146 
.909055 
.908964 
.908873 

1^52 
1.52 
1.52 
1.53 

.857803 
.858069 
.858336 
.858803 

4^43 
4.45 
4.43 
4.43 

.142197 
.141931 
.141664 
.141398 

13 

12 
11 
10 

51 
52 
53 
54 
55 
56 

9.767649 

.767824 
.767999 
.768173 

.768348 
.768522 

2.92 
2.92 
2.90 
2.92 
2.90 

9.908781 
.908690 
.908599 
.908507 
.908416 
.90832-1 

1.52 
1.52 
1.53 
1.52 
1.53 

9.858868 
.859134 
.859400 
.859666 
.859982 
.800198 

4.43 
4.43 
4.43 
4.43 
4.43 
4  4^ 

10.141132 
.140800 
.140600 
.140334 
.140008 
.139802 

9 

8 

6 
5 

4 

57 

58 
59 
60 

.768697 
.768871 
.769045 
9.769219 

2^90 
2.90 
2.90 

.908333 
.908141 
.908049 
9.907958 

1.53 
1.53 
1.52 

.860464. 
.860730 
.860995 
9.861261 

4.'43 
4.42 
4.43 

.139536 
.139270 
.139005 
10.138739 

3 
2 
1 
0 

' 

Cosine.  I  D.  1".  il   Sine. 

D.  1". 

Cotang.  D.  i".  i  Tang.    ' 

125 


394 


COSINES,  TANGENTS,   AND  COTANGENTS. 


143' 


t 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 
1 
2 

9.769219 
.769393 
.769566 

2.90 

2.88 

9.907'958 
.907866 
.907774 

.53 
.53 

pro 

9.861261 
.861527 
.861792 

4.43 
4.42 

4,|O 

10.138739 
.138473 

.138208 

60 
59 

58 

3 

.769740 

2.90 

.907682 

.Do 

Mk 

.862058 

.4o 

.137942 

57 

4 
5 

.769913 

.770087 

2.88 
2.90 

2QO 

.907590 
.907498 

.OO 

.53 

tQ 

.862323 

.862589 

4  '.43 

.137677 
.137411 

56 
55 

6 

7 

.770260 
.770433 

.OO 

2.88 

O  OO 

.907406 
.907314 

.OO 

.53 

eo 

.862854 
.863119 

4'.42 

4/1Q 

.137146 
.136881 

54 

53 

8 
9 
10 

.770608 
.770779 
.770952 

2.oo 
2.88 
2.88 
2.88 

.907222 
.907129 
.907037 

.Do 
.55 
1.53 
1.53 

.863385 
.863650 
.863915 

.4o 

4.42 
4.42 
4.43 

.136615 
.136350 
.136085 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.771125 
.771298 
.171470 

.771643 
.771815 
.771987 
.772159 
.772331 
.772503 
.772675 

2.88 

2.87 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 

9.906945 
.906&52 
.9067'60 
.906667 
.906575 
.906482 
.906389 
.906296 
.906204 
.906111 

1.55 
1.53 
1.55 
1.53 
1.55 
1.55 
1.55 
1.53 
1.55 
1.55 

9.864180 
.864445 
.864710 
.864975 
.865240 
.865505 
.865770 
.866035 
.866300 
.866564 

4.42 
4.42 
4.42 
4.42 
4.42 
'  4.42 
4.42 
4.42 
4.40 
4.43 

10.135820 
.i:»555 
.135290 
.135025 
.1:34760 
.134495 
.134230 
.1:33965 
.133700 
.133436 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.772847 
.773018 
.773190 
.773361 
.773533 
!  778704 
.773875 
.774046 
.774217 
.774388 

2.85 
2.87 
2.85 
2.87 
2.85 
2.85 
2.85 
2.85 
2.85 
2.83 

9.906018 
.905925 
.905832 
.905739 
.905645 
.905552 
.905459 
.905366 
.905272 
.905179 

1.55 
1.55 
1.55 
1.57 
1.55 
1.55 
1.55 
1.57 
1.55 
1.57 

9.866829 
.867094 
.867358 
.867623 

.867887 
.868152 
.868416 
.868680 
.868945 
.869209 

4.42 
4.40 
4.42 
4.40 
4.42 
4.40 
4.40 
4  42 
4.40 
4.40 

10.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131584 
.131320 
.131055 
.130791 

39 

38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 

9.774558 
.774729 

2.05 

9.905085 
.904992 

1.55 

9.869473 
.86,7737 

4.40 

10.130527 
.130263 

29 

28 

33 
34 
35 

.774899 
.775070 
.775240 

2.83 
2.85 
2.83 

2QQ 

.904898 
.904804 
.904711 

1  .57 
1.57 
1.55 

Itr* 

.87'0001 
.870265 
.870529 

4.40 
4.40 
4.40 
4  /in 

.129999 
.129785 
.129471 

27 
26 
25 

36 
37 

.775410 

.775580 

.00 
2.83 

.904617 
.904523 

.  D* 

1.57 

.870793 
.871057 

.40 
4.40 

.129207 
.  128943 

24 
23 

38 

.775750 

2.83 

2  CO 

.904429 

1  .57 

.871321 

4.40- 

4Af\ 

.  128679 

22 

39 
40 

.775920 
.776090 

.00 
2.83 
2.82_ 

.904335 
.904241 

1  .57 
1.57 
1.57 

.871585 
.871849 

.40 

4.40 
4.38 

.128415 
.128151 

21 

20 

41 

9.776259 

2QO 

9.904147 

1  57 

1  9.872112 

44f\ 

10.127888 

19 

42 

.776429 

.OO 

.904053 

IKty 

.872376 

:.4U 

4A(\ 

.127624 

18 

43 
44 

45 

.776598 
.776768 
.776937 

2^83 

2.82 

.903959 
.903864 
.903770 

.Df 

1.58 
1.57 

.872640 
.872903 
.873167 

.40 
4.38 
4.40 

.127360 
.127097 
.126833 

17 
16 
15 

46 

.777106 

2.82 

.903676 

1  .57 

.873430 

4.38 

.126570 

14 

47 

.777275 

2.82 

200 

.903581 

1  .58 

1frf*f 

.873694 

4.40 

4QQ   * 

.126306 

13 

48 

.777444 

.0,4 

.903487 

.Of 

.873957 

.00 

.126043 

12 

49 
50 

.777613 
.777781 

2.82 
2.80 
2.82 

.903392 
.903298 

1.57 
1.58. 

.874220 

.874484 

4.38 
4.40 
4.38 

.125780 
.125516 

11 
10 

51 

9.777950 

9.903203 

Ip-Q 

9.874747 

400 

10.125253 

9 

52 

.778119 

2.  o3 

.903108 

.08 

IfcM 

.875010 

.00 

.124990 

8 

53 
54 
55 
56 
57 

.778287 
.778455 
.778624 
.77'8792 
.778960 

2!80 
2.82 
2.80 
2.80 

O  QA 

.903014 
.902919 
.902824 
.902729 
.902634 

.Of 

1.58 
.58 
.58 
.58  . 

.875273 
.875537 
.875800 
.876063 
.876326 

4.38 
4.40 
4.38 
4.38 
4.38 

.124727 
.124463 
.124200 
.123937 
.123674 

7 
6 
5 
4 
3 

58 

.779128 

/w.OU 

.902539 

.58 

.876589 

4.38 

.123411 

2 

59 
60 

.779295 
9.779463 

2.78 
2.80 

.902444 
9.902349 

.58 
1.58 

.876852 
9.877114 

4.38 
4.37 

.123148 
10.122886 

1 
0 

* 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.r. 

Tang. 

' 

395 


53° 


37C 


TABLE  XXV. -LOGARITHMIC  SINES, 


, 

Sine. 

D.  1". 

Cosine. 

D.  1'.  i   Tang. 

D.  1". 

Cotang.  |  ' 

'•' 

1        1 

0 

I 

3 
4 
5 

6 

7 
8 

9.779463 
.779631 
.779798 
.779966 
.780133 
.780300 
.780467 
.780334 
.780801 

2.80 

2.78 
2.80 
2.78 
2.78 
2.78 
2.78 
2.78 

2r-o 

9.902349 
.902253 
.902158 
.902063 
.901967 
.901872 
.901776 
.901681 
.901585 

1.60 
1.58 
1.58 
1.60 
1.58 
1.60 
1.58 
1.60 

]eo 

9.&77114 

.877377 
.877640 
.8771:03 
.878165 
.878428 
.878691 
.878953 
.879216 

4.38 
4.38 
4.38 
4.37 
4.38 
4.38 
4.37 
4.38 

A  Q7 

10.122886 
.122623 

.122360 
.122097 
.121835 
.121572 
.121809 
.121047 
.120784 

60 
59 
58 
57 
56 
55  . 
54 
53 
52 

9 

.780968 

.  (8 

.901490 

.OO 
1AA 

.879478 

4  .  Gi 

4OQ 

.  120522 

51 

10 

.781134 

2.77 
2.78 

.901394 

.00 

1.60 

.879741 

.GO 

4.37 

.120259 

50 

11 
12 

9.7'81301 
.781468 

2.78 

9.901298 
.901202 

1.60 

9.880003 
.880265 

4.37 

4OQ 

10.119997 
.119735 

49 
48 

13 
14 

.781634 
.781800 

2.77 
2.77 

.901106 
.901010 

140 

.880528 
.8H0790 

.GO 

4.37 
4  S7* 

.119472 
.119210 

47 
46 

15 
16 

.781966 
.782133 

2.77 

2.77 

.900914 

.900818 

140 

i  .881052 
.881314 

4^37 

A  qo 

.118948 
.118686 

45 
44 

17 

18 

.782298 
.782464 

2.77 

2.77 

.900722 
.900626 

I'M 

1AO 

.881577 
.881839 

4  .  oo 

4.37 

4Q*7 

.118423 
.118161 

43 
42 

19 

.782630 

2-11 

.900529 

.TO 
1AA 

.882101 

.04 

A  Q7 

.117895)  41 

20 

.782796 

2.  it 
2.75 

.900433 

.  DU 

1.60 

.882363 

4  .  O* 

4.37 

.117637 

40 

21 

OO 

23 

9.782961 
.783127 
.783292 

2.77 
2.75 

9.900337 
.900240 
.900144 

1.62 
1.60 

9.882625 

.882887 
.883148 

4.37 
4.35 

40.7 

10.117375 
.117113 

.116852 

39 
38 
37 

24 
25 

26 

27 

.783458 
.783623 
.783788 
.783953 

2.77 
2.75 
2.75 
2.75 

.900047 
.899951 
.899854 
.899757 

1:00 

1.62 
1.62 

.883410 
.883672 
.883934 
.884196 

.G< 

4.37 
4.37 
4.37 

4  OK 

.116590 
.116328 

.1160(56 
.115804 

36 
35 
34 
33 

28 
29 

.784118 

.784282 

2.75 
2.73 

.899660 
.899564 

140 

.884457 
.884719 

.GO 

4.37 

A  OK 

.115543 

.115281 

32 
31 

30 

.784447 

2.75 
2.75 

.899467 

148 

.884980 

4.  GO 

4.37 

.115020 

30 

31 
32 
33 
34 

9.784612 
.784776 
.784941 
.785105 

2.73 
2.75 

2.73 

9.899370 
.899273 
.899176 
.899078 

1.62 
1.62 
1.63 

1AO 

9.885242 
.885504 
.885765 
.886026 

4.37 

4.35 
4.35 

A  Q7 

10.114758 
.114496 
.114235 
.113974 

29 
28 
27 
26 

35 

36 
37 
38 
39 
40 

.785269 
.785433 
.785597 
.785761 
.785925 
.786089 

2^73 
2.73 
2.73 
2.73 
2.73 
2.7'2 

.898981 
.898884 
.898787 
.898689 
.898592 
.898494 

.0x5 

1.62 
1.62 
1.63 
1.62 
1.63 
1.62 

.886288 
i  .886549 
.886811 
1  .887072 
.887883 
.887594 

4.o< 

4.35 
4.37 
4.35 
4.35 
4.35 
4.35 

.113712 
.113451 

.113189 
.  .112928 
.112667 
.112406 

25 
24 
23 

22 

!  21 

20 

41 
42 
43 
44 
45 
46 
47 
48 
49 

9.786252 

.786416 
.786579 
.786742 
.786906 
.787069 
.787232 
.787395 
.787557 

2.73 

2.72 
2.72 
2.73 
2.72 
2.72 
2.72 
2.70 

9.898397 
.898299 
.898202 
.898104 
.898006 
.897908 
.807810 
.897712 
.897614 

1.63 
1.62 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 

!  9.887855 
i  .888116 
.888378 
.888639 
.888900 
.889161 
.889421 
.889682 
.889943 

4.35 
4.37 
4.35 
4.35 
4.35 
4.33 
4.35 
4.35 

4O-r 

10.112145  |  19 
.111H84  ;  18 
.111(522  I  17 
.111361   16 
.111100   15 
.110839  !  14 
.110579  j  13 
.110318   12 
.110057   11 

50 

.787720 

2.72 

2.72 

.897516 

1  .63 
1.68 

.890204 

.  y> 
4.35 

.109796   10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
GO 

9.787883 
.788045 
.788208 
.788370 
.788532 
.788694 
.788856 
.789018 
.789180 
9.789342 

2.70 

2.72 
2.70 
2.70 
2.70 
2.70 
2.70 
2.70 
2.70 

9.897418 
.897320 
.897222 
,897123 
.897025 
.896926 
.896828 
.896729 
.896631 
9.896532 

1.63 
1.63 
1.65 
1.63 
1.65 
1.63 
1.65 
1.63 
1.65 

9.890465 

.890725 
.890986 
.891247 
.891507 
.89176S 
.892028 
.892289 
.892549 
9.892810 

4.33 
4.35 
4.35 
4.33 
4.35 
4.33 
4.35 
4.33 
4.35 

10.109535 
.109275 
.109014 
.108753 
.108493 
.106282 
.107972 
.107711 
.107451 
10.107190 

9 

!  8 

6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D  1".  il  Sine,  i  D.  1°. 

!  Cotang.   D.  1".   Tang,   i  ' 

127C 


COSINES,  TANGENTS,   AND  COTANGENTS. 


141' 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

/ 

0 

9.789342 
.789504 

2.70 

9.896532 
.890433 

1.65 
1  r*3 

9.892810 
.893070 

4.33 

4  QC 

10.107190 
.106930 

60 
59 

2 
3 

.789665 
.789827 

2.68 

2.70 

2AQ 

.896335 
.896236 

.  Do 

1.65 

1/»" 

.893331 
.89:3591 

.OO 

4.33 

4OQ 

.106669 
.106409 

58 
57 

4 

.789988 

.DO 

.896137 

.  DO 

.893851 

.OO 

400 

.106149 

56 

5 

.790149 

2.08 

2AQ 

.896038 

1  .65 
COL 

.894111 

.OO 

4oe 

.105889 

55 

6 

.790310 

.DO 

2/»o 

.895939 

.  DO 

AX   '• 

.894372 

.  oO 

400 

.105628 

54 

.790171 

.OO 
2AQ 

.895840 

.DO   ; 

ftrr 

.894632 

.OO 
A  QQ 

.105368 

53 

8 

.790632 

.Do 
2AQ 

.895741 

.DO 

.894892 

4.Oi* 

4QO 

.105108 

52 

9 

.790793 

.DO 

.895641 

.67    ; 

.895152 

.00 

4QQ 

.104848 

51 

10 

.790954 

2.68 

2.68 

.895542 

!  .05 
.65  i 

.895412 

.00 

4.33 

.10-1588 

50 

11 
12 
13 
14 
15 
16 

9.791115 
.791275 
.791436 
.791596 
.791757 
.791917 

2.67 
2.68 
2.67 
2.68 

2.67 

9.895443 
.895343 
.895244 
.895145 
.895045 
.894945 

.67  : 
.65 
.65 
.67-  ! 
.67 

9.895672 
.895932 
.896192 
.898452 
.896712 
.898971 

4.33 
4.33 
4.33 
4.33 
4.32 
400 

10.104328 
.104068 
.103808 
.103548 
.103288 
.103029 

49 

48 
47 
46 
45 
44 

17 
18 
19 

20 

.792077 
.792237 
.792397 

.792557 

2.67 
2.67 
2.67 
2.67 
2.65 

.894846 
.894746 
.894646 
.894546 

.65 

.67  ! 
.67  i 
.67  ; 
.67 

.897231 
.897491 
.897751 
.898010 

.00 
4.33 
4.33 
4.32 
4.33 

.102769 
.102509 
.102249 
.101990 

43 
42 
41 
40 

21 
22 
23 

24 
25 

26 
27 

9.792716 
.793376 
.793033 

.793195 
.793354 
.79S514 
.793673 

2.67 
2.65 
2.67 
2.65 
2.67 
2.65 

9.894446 
.894346 
.894246 
.894146 
.894046 
.893946 
.893846 

.67 
.67 
.67 
.67 
.67 
.67 

/>Q 

9.898270 
.898530 
.898789 
.899049 
.899308 
.899568 
.899827 

4.33 

4.32 

4.  as 

4.32 
4.33 
4.32 
400 

10.101730 
.101470 
.101211 
.100951 
.100692 
.100432 
.100173 

39 
38 
37 
36 
35 
34 
33 

28 
29 

.793832 
.793991 

2.65 

2.65 
o  /»~ 

.893745 
.893645 

.DO 

.67 

f»Q 

.900087 
.900346 

.00 
4.32 

4-  QO 

.099913 
.099654 

32 

31 

30 

.794150 

«  .  uo 
2.63 

.893544 

.OO 

.67 

.900605 

.34 

4.32 

.099395 

30 

31 

9.794303 

2   n* 

9.893444 

AQ 

9.900864 

400 

10.099136 

29 

32 

.794467 

.DO 
o  /'X. 

.893343 

.DO 
A*7 

.901124 

.00 

40.) 

.098876 

2S 

33 

.794G26 

/&.DO 
2  OS 

.893243 

.Di 
AQ 

.901383 

,oSs 

4na 

.098617 

27 

34 

.794784 

.00 

.893142 

.DO 
f*Q 

.901642 

.O^ 

.098358 

26 

,35 

.794942 

2.63 

2A^ 

.893041 

.DO 
ro 

.901901 

4.32 
400 

.098099 

25 

36 

.795101 

.DO 

.892940 

.  vO 

f»Q 

.902160 

.  -  *  - 

4QQ 

.097840 

24 

37 

.795259 

2.63 

2£>q 

.892839 

.DO 
A*7 

.902420 

.00 

4QO 

.097580 

23 

38 

.795417 

.  DO 
O  AQ 

.892739 

.D( 
AQ 

.902679 

.O« 

4QO 

.097321 

22 

39 

40 

.795575 

.795733 

/«.  Do 

2.63 
2.63 

.892638 
.892536 

.DO 

.70 
.68 

.902938 

.903197 

.0^ 

4.32 
4.32 

.097062 
.096803 

21 
20 

41 

9.795891 

2fiO 

9.892435 

AQ 

9.903456 

4OA 

10.096544 

19 

42 

.796049 

.DO 

2  A3 

.892334 

.DO 
AQ 

.903714 

.OV 

4QO 

.096286 

18 

43 

.796205 

.O^ 
2/»o 

.892233 

.DO 

00 

.903973 

•CEQ 

.096027 

17 

44 

.796364 

.DO 

21'.) 

.892132 

.DO 
r-rt 

.904232 

4.32 

4QO 

.095768 

16 

45 

.796521 

.  D.4 
t>  no 

.892030 

.  rfU 

AQ 

.904491 

.O/v 

400 

.095509 

15 

40 

.796679 

^.DO 

2j'.> 

.891929 

.DO 

.  904750 

.06 
4OA 

.095250 

14 

47 

.796836 

.  '  )  •- 
O  AO 

.891827 

/»Q 

.905008 

.oU 

4QO  * 

.094992 

13 

48 

.796993 

<£  .  o/£ 

.891726 

.DO 

.905267 

.0*5 
4QO 

.0947:33 

12 

49 

.797150 

O  AO 

.891624 

.70 

AQ 

.905526 

.64 

400 

.004474 

11 

50 

.797307 

A,  U-J 

2.02 

.891523 

.DO 

.70 

.905785 

blSI 

4.30 

.094215 

10 

51 

9.797464 

O  AO 

9.891421 

9.906043 

4QO 

10.093957 

9 

52 
53 
54 
55 
56 
57 
58 

.797621 
.797777. 

.797934 
.798091 
.798247 
.798403 
.798560 

f\i  .  \)<i 

2.60 
2.62 
2.62 
2.60 
2.60 
2.62 

2nf\ 

.891319 
.891217 
.891115 
.891013 
.890911 
.890809 
.890707 

.70 
.70 
.70 
.70 
.70 
1.70 

.906302 
.906.5(50 
.906819 
.907077 
.907336 
.907594 
.907853 

•  t&9 

4.30 

4.32 
4.30 
4.32 
4.30 
4.32 

.093698 
.093440 
.093181 
.092923 
.092664 
.092406 
.092147 

8 

6 
5 

4 
3 

2 

59 
60 

.798716 
9.798872 

.6U 
2.60 

.890605 
9.890503 

1  .70 
1.70 

.908111 
9.908369 

4.30 
4.30 

.091889 
10.091631 

1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1".  i 

Cotang. 

D.  1". 

Tang. 

' 

-8,97 


39° 


TABLE  XXV. -LOGARITHMIC  SINES, 


1 

' 

Sine. 

D.  1". 

Cosine. 

D.I". 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

2 
3 

4 
5 
6 

7 

9.798872 
.799028 
.799184 
.799,339 
.799495 
.799651 
.799806 
.799962 

2.60 
2.60 
2.58 
2.60 
2.60 
2.58 
2.60 

9.890503 
.890400 
.890298 
.890195 
.890093 
.889990 
.889888 
.889785 

1.72 
1.70 
1.72 
.70 

.72 
1.7'0 

:  .72 

9.908369 
.908628 
.908886 
.909144 
.909402 
.909660 
.909918 
.910177 

4.32 

4.30 
4.30 
4.30 
4.30 
4.30 
4.32 

10.091631 
.09137'2 
.091114 
.090856 
.090598 
.090340 
.090082 
.089823 

60 

59 
58 
57 
56 
55 
54 
53 

8 
9 
10 

.800117 
.800272 
.800427 

2^58 
2.58 
2.58 

.889682 
.889579 
.889477 

1.72 
1.70 
1.72 

.910435 
.910693 
.910951 

4^30 
4.30 
4.30 

.089565 
.089307 
.089049 

52 
51 
50 

11 

9.800582 

2trQ 

9.889374 

9.911209 

10.088791 

49 

12 

13 

.800737 
.800892 

.Do 

2.58 

.889271 
.889168 

1  .72 
1.72 

.911467 
.911725 

4^30 

.088533 

.088275 

48 
47 

14 

15 
16 

.801047 
.801201 
.801356 

2.58 
2.57 

2.58 

.889064 
.888961 
.888858 

:  '.72 

.72 

.911982 
.912240 
.912498 

4.28 
4.30 
4.30 

.088018 
.087760 
.087502 

46 
45 
44 

17 
18 
19 
20 

.801511 
.801665 
.801819 
.801973 

2.58 
2.57 
2.57 
2.57 
2.58 

.888755 
.888651 
.888548 
.888444 

'.  .72 
.73 
.72 
.73 

.72 

.912756 
.913014 
.913271 
.913529 

4.30 
4.30 

^4.28 
4.30 
4.30 

.087244 
.086986 
.086729 
.086471 

43 

42 
41 
40 

21 

9.802128 

9.888341 

9.913787 

10.086213 

39 

22 

.802282 

2.57 

.888237 

'.  .73 

.914044 

4.28 

.085956 

38 

23 
24 

.802436 
.802589 

2.57 
2.55 

2KT* 

.888134 
.888030 

.72 
.73 
iyo 

.914302 
.914560 

4^30 

.085698 
.085440 

37 

36 

25 
26 

.802743 

.802897 

.57 

2.57 

.887926 
.887822 

.  .  <o 
.73 

r-o 

.914817 
.915075 

4.30 

.085183 
.084925 

34 

27 
28 
29 

.803050 
.803204 
.803357 

2  .  55 
2.57 
2.55 

.887718 
.887614 
.887510 

.  .  to 
.73 

.73 

.915332 
.915590 
.915847 

4.  'SO 

4.28 

4OQ 

.084668 
.084410 
.084153 

32 
31 

30 

.803511 

2  .  .r-7 
2.55 

.887406 

.73 
.73 

.916104 

.Iso 
4.30 

.083896 

30 

31 
32 

9.803664 

.803817 

2.55 

9.887302 

.887198 

.73 

9.916362 
.916619 

4.28 

10.083638 

.083381 

29 

28 

33 
34 

.803970 
.804123 

2.55 
2.55 

.887093 
.886989 

.75 
.73 

.916877 
.9171:54 

4  '.28 

4"  OQ 

.083123 

.082866 

27 
26 

35 

.804276 

2.55 

.886885 

•  ''2 

.917391 

.ISO 

.082609 

25 

36 

.804428 

2.55 

.886780 

.75 

r-o 

.917648 

4.28 

.082-352 

24 

37 

38 
39 

.804581 
.804734 

.804886 

2.55 
2.55 
2.53 

.886676 
.886571 
.886466 

.  IO 

.75 

•£5 

.917906 
.918163 
.918420 

4~28 
4.28 

4OQ 

.082094 
.081837 
.081580 

23 

22 
21 

40 

.805039 

2.55 
2.53 

.886362 

'.75 

.918677 

.x/O 

4.28 

.081323 

20 

41 
42 

9.805191 
.805343 

2.53 

9.886257 
.886152 

.75 

9.918934 
.919191 

4.28 

10.081066 

.080809 

19 
18 

43 
44 

.805495 
.805647 

2.53 
2.53 

.886047 
.885942 

.75 
.75 

.919448 
.919705 

4.28 

4.28 

.080552 
.080295 

17 
16 

45 
46 

47 
48 
49 

.805799 
.805951 
.806103 
.806254 
.806406 

2.53 
2.53 
2.53 
2.52 
2.53 

.885837 
.885732 
.885627 
.885522 
.885416 

.75 
.75 
.75 

.75 

.77 

.919962 
.920219 
.920476 
.920733 
.920990 

4^28 
4.28 
4.28 
4.28 

4OQ 

.080038 
.079781 
.079524 
.079267 
.079010 

15 
14 
13 
12 
11 

50 

.806557 

2.52 
2.53 

!  .885311 

.  77> 

.77 

.921247 

•.so 
4.27 

.078753 

10 

51 
52 

9.  -806709 
.806860 

2.52 

9.885205 
.885100 

.75 

9.921503 

.921760 

4.28 

10.078497 
.078240 

9 

8 

53 

.807011 

2.52 

.884994 

'  r"~ 

.922017 

a  Is 

.077983 

7 

54 

.807163 

2.53 

.884889 

'X~ 

.922274 

4  27 

.077726 

6 

55 

.807314 

2.52 

.884783 

.  l  1 

.922530 

.077470 

5 

56 

.807465 

2.52 

.884677 

.77 

.922787 

4.28 

.077213 

4 

57 
58 

.807615 
.807766 

2.50 
2.52 

.884572 
.884466 

•I7 

.923044 
.923300 

4^27 

4OQ 

.076956 
.076700 

3 

2 

59 

.807917 

A.oA 

.884360 

.  77 

.923557 

.  ^5o 
4OQ 

.076443 

1 

60 

9.808067 

2.50 

9.884254 

.77 

9.923814 

.**& 

10.076186 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  r. 

1  Cotang. 

D.  r. 

Tang. 

' 

129° 


303 


COSINES,  TANGENTS,  AND  COTANGENTS. 


139C 


* 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  1'. 

Cotang. 

'- 

0 

1 

9.808067 
.808218 

2.52 

9.884254 

.884148 

:  .77 

9.923814 
.924070 

4.27 

10.076186 
.075930 

60 

59 

2 
3 
4 

5 
6 
7 
8 
9 
10 

.808:368 
.808519 
.808669 
.808819 
.808960 
.809119 
.809269 
.809419 
.809569 

2  '.52 
2.50 
2.50 
2.50  i 
2.50  1 
2.50 
2.50 
2.50 
2.48 

.884042 
.883936 
.883829 
.883723 
.883617 
.883510 
.883404 
.883297 
.883191 

'.77 

.78 
.77 
.77 
.78 
.77 
.78 
.77 
.78 

.924327 
.924583 
.924840 
.925096 
.925352 
.925609 
.925865 
.926122 
.926378 

4'.  27 
4.28 
4.27 
4.27 
4.28 
4.27 
4.28 
4.27 
4.27 

.075673 
.075417 
.075160 
.074904 
.074648 
.074391 
.074135 
.073878 
.073622 

58 
57 
56 
55 
54 
53 
52 
51 
58 

11 
12 

9.809718 
.809868 

2.50 
2/itt 

9.883084 
.882977 

.78 

9.926634 
.926890 

4.27 

4OQ 

10.07a366 
.073110 

49 

48 

13 

.810017 

.4o 

.882871 

'.  .77 

.927147 

.<Jo 

.072853 

47 

14 
15 
16 
17 
18 

.810167 
.810:316 
.810465 
.810614 
.810763 

2.50 

2.48 
2.48 
2.48 
2.48 

2.Q 

.882764 
.882657 
.882550 
.882443 
.882386 

.78 
.78 
.78 
.78 
.78 

.927403 
.927659 
.927915 
.928171 

.928427 

4.27 
4.27 
4.27 
4.27 
4.27 

.072597, 
.072341 
.072085 
.071829 
.071573 

46 
45 
44 
43 
42 

19 
20 

.810912 
.811061 

.4o 
2.48 
2.48 

.882229 
.882121 

.78 
1.80 
1.78 

.928684 
.628940 

4.28 
4.27 
4.27 

.071316 
.071060 

41 
40 

21 
22 

9.811210 

.811358 

2.47 

9.882014 
.881907 

1.78 

9.929196 
.929452 

4.27 

10.070804 
.070548 

39 

38 

23 
24 
25 

26 

.811507 
.811655 
.811804 
.811952 

2.48 
2.47 
2.48 
2.47 

.881799 
.881692 
.881584 
.881477 

1  .80 
1.78 
1.80 
1.78 

1QA 

.929708 
.929964 
.930220 
.930475 

4.27 
4.27 
4.27 
4.25 
4  try 

.070292 
.070036 
.069780 
.069525 

37 
36 

34 

27 
28 
29 
30 

.812100 
.812248 
.812396 
.812544 

2.47 

2.47 
2.47 
2.47 
2.47 

.881369 
.881261 
.881153 
.881046 

.oU 
1.80 
1.80 
1.78 
1.80 

.930731 
.930987 
.931243 
.931499 

:.xf 

4.27 
4.27 
4.27 
4.27 

.069269 
.069013 
.068757 
.068501 

33 
32 
31 
30 

31 
32 

as 
at 

35 
36 
37 

38 

9.812692 
.812840 
.812988 
.813135 
.813283 
.81:3430 
.813578 
.8137'25 

2.47 
2.47 
2.45 
2.47 
2.45 
2.47 
2.45 

2  AX* 

9.880938 
.880830 
.880722 
.880613 
.880505 
.880397 
.880289 
.880180 

1.80 
1.80 
1.82 
1.80 
1.80 
1.80 
1.82 

9.931755 
.932010 
.932266 
.932522 
.932778 
.933033 
.933289 
.933545 

4.25 
4.27 
4.27 
4.27 
4.25 
4.27 
4.27 

4e\K 

10.068245 
.067990 
.067734 
.067478 
.067222 
.066967 
.066711 
.066455 

29 
28 
27 
26 
25 
24 
23 
22 

39 
40 

.813872 
.814019 

.4o 
2.45 
2.45 

.880072 
.879963 

1.82 
1.80 

.933800 
.934056 

.flu 

4.27 
4.25 

.066200 
.065944 

21 
20 

41 
42 

9.814166 
.814313 

2.45 

9.879855 
.879746 

1.82 

9.934311 
.934567 

4.27 

10.065689 
.065433 

19 
18 

43 
44 

45 
46 
47 
48 
49 
50 

.814460 
.814607 
.814753 
.814900 
.815046 
.815193 
.815:339 
.815485 

2.45 
2.45 
2.43 
2.45 
2.43 
2.45 
2.43 
2.43 
2.45 

.879637 
.879529 
.879420 
.879311 

.879202 
.879093 
.878984 
.878875 

1  .82 
1.80 
1.82 
1.82 
1.82 
1.82 
1.82 
1.82 
1.82 

.934822 
.935078 
.9a5333 
.985589 
.935844 
.936100 
.936355 
.936611 

4.25 
4.27 
4.25 
4.27 
4.25. 
4.27 
4.25 
4.27 
4.25 

.065178 
.064922 
.064667 
.064411 
.064156 
.063900 
.063645 
.063389 

17 
16 
15 
14 
13 
12 
11 
10 

51 
52 

9.815632 

.815778 

2.43 

2  An 

9.878766 

.878656 

1.88 

9.936866 
.937121 

4.25 

10.063134 
.062879 

9 

8 

53 

54 

.815924 
.816069 

.4o 

2.42 

.878547 
.8784:38 

1.82 
1.82 

.937377 
.937632 

4.27 
4.25 

.062623 
.062368 

7 
6 

55 
56 

57 

.816215 
.81  6361 
.816507 

i  2^43 
2.43 

.878328 
.878219 
.878109 

1  .83 
1.82 
1.83 

1OO 

.937887 
.938142 
.938398 

4.25 
4.25 

4.27 

.062113 
.061858 
.061602 

5 
4 
3 

58 
59 

.816652 
.816798 

2^43 

.877999 

.877890 

.00 

1.82 

.938653 
.938908 

4.25 
4.25 

.061347 
.061092 

2 
1 

60 

9.816943 

2.42 

9.877780 

1.83 

9.939163 

4.25 

10.060837 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.  1". 

Tang. 

' 

130° 


399 


49° 


TABLE  XXV. -LOGARITHMIC  SINES, 


138C 


' 

Sine. 

D.  r. 

i 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.816943 

9.877780 

1QO 

9.939163 

10.060837  60 

1 

2 

.817088 
.817233 

2.42 
2.42 

.877670 

.877560 

.OO 

1.83 

100 

.939418 
.939673 

4  '.25 

.060582  59 
.060327  !  58 

3 
4 

.817379 
.817524 

2.43 

2.42 

24f\ 

.877450 
.877340 

.00 
1.83 

1QQ 

.939928 
.940183 

4^25 

.060072  !  57 
.059817  !  56 

5 
6 

.817668 
.817813 

.40 

2.42 

2AC\ 

.877*30 
.877120 

.OO 

1.83 

100 

.940439 
.940694 

4.27 
4.25 
4  ox 

.059561  i  55 
.059306  54 

7 
8 
9 

.817958 
.818103 
.818247 

.4/4 

2.42 

2.40 

.877010 
.876893 
.876789 

.00 
1.85 
1.88 

1Q*= 

.940949 
.941204 
.941459 

.&> 
4.25 
4.25 

.059051 
.058796 
.058541 

53 

52 
51 

K) 

.818392 

2^40 

.876678 

.oO 

1.83 

.941713 

4^25 

.058287 

50 

11 

9.818536 

.818681 

2.42 

9.876568 
.876457 

1.85 

i   OO 

9.941968 
.942223 

4.25 

10.058032 
.057777 

49 
48 

13 
14 

.818825 
.818989 

2.40 
2.40 

.876347  i  f'S2 
.876236   ?-S 

.942478 
.942733 

4.25 
4.25 

.057522 
.057267 

47 

46 

15 
16 
17 

.819113 
.819257 
.819401 

2.40 
2.40 
2.40 

2A(\ 

.876125 
.876014 
.875904 

1  .Oi> 

1.85 
1.83 

1QX 

.942988 
|  .943243 
!  .943498 

4.25 
4.25 

4.25 

4OQ 

.057012 
.056757 
.056502 

45 
44 
43 

18 
19 

20 

.819515 
.819889 
.819332 

.4U 

2.40 
2.33 
2.40 

.875793 

.87568-2 
.875571 

.OO 

1.85 
1.85 
1.87 

i  .943752 
!  .944007 
I  .944262 

.  -•> 

4-.  25 
4.25 
4.25 

.056248 
.055993 
.055738 

42 
41 
40 

21 
22 

9.819976 

.820120 

2.40 

2OQ 

9.875459 

.875348 

1.85 

9.944517 
.944771 

4.23 

10.055483 

.055229 

39 
38 

23 

.820233 

.OO 

.875237 

1  C" 

.945026 

**'~~ 

.054974 

37 

24 
25 

26 

.820408 
.820550 
.820893 

2^40 
2.33 

2QQ 

.875126 
.875014 
.874903 

1  .00 
1.87 
1.85 

.945281 
.945535 
.945790 

4  .  2o 
4.23 
4.25 

.054719 
.054465 
.054210 

36 
35 
34 

27 

.820836 

.  OO 
aOO 

.874791 

Q- 

.946045 

A  OQ 

.058955 

33 

23 
29 

.820979 
.821122 

.OO 

2.38 

2QQ 

.874680 
.874568 

1.87 

.946299 
.946554 

4*25 

.053701  i  32 
.053446  !  31 

30 

.821265 

.00 
2.37 

.874456 

1.87 

.946808 

4  "25 

.053192 

30 

31 
32 
33 
34 

9.8-21407 
.821550 
.8-21693 
.8218'35 

2.38 
2.38 
2.37 

o  07 

9.874344 
.874232 
.874121 

.874009 

1.87 
1.85 
1.87 

100 

9.947063 
.947318 
.947572 
.947827 

4.25 
4.23 
4.25 

10.052937 
.052682 
.052428 
.052173 

29 
28 

27 
26 

35 
36 
37 
38 
39 
40 

.•821977 
.822120 
.822262 
.822404 
.823546 
.822688 

2^38 
2.37 
2.37 
2.37 
2.37 
2.37 

.873896 
.873784 
.873672 
.873560 
.873448 
.873335 

.00 

1.87 
1.87 
1.87 
1.87 
1.88 
1.87 

.948081 
.948335 
.948590 
.948844 
.949099 
.949353 

4/23 
4.25 
4.23 
4.25 
4.23 
4.25 

.051919 
.051665 
.051410 
.051156 
.050901 
.050647 

25 
24 

22 

21 
20 

41 
42 
43 
44 
45 

9.822330 
.822972 
.823114 
.823255 
.8-23397 

2.37 
2.37 
2.35 
2.37 

9.873223 
.873110 

.872998 
.872885 
.872772 

1.88 

1.87 
1.88 
1.88 

1QQ 

9.949608 
.949862 
.950116 
.950371 
.950625 

4.23 
4.23 
4.25 
4.23 

4QO 

10.050392 
.050K38 
.049884 
.049629 
.049375 

19 
18 
17 
16 
15 

46 

.823539 

.4.67 

.872659 

.OO 

IQrt 

.950879 

.  £3 
4OQ 

.049121  1  14 

47 

.823680 

2.35 

.872547 

.of 

100 

.951133 

.&) 

.048867  1  13 

48 

.823821 

2.35 

20? 

.872434 

.00 

100 

.951388 

4.25 

.048612   12 

49 

.823963 

.of 

O  QK. 

.872321 

.00 
1  ftH 

.951642 

I'M 

.048858  i  11 

50 

.824104 

,«.OO 

2.35 

.872208 

l.« 

.951896 

4:23 

.018104   10 

51 
52 

9.824245 

.824386 

2.35 

i  9.872095 

!  .871981 

1.90 

-,   00 

9.952150 
.952405 

4.25 

10.047850 
.047595 

9 

8 

53 
54 

.824527 
.824668 

2.35 
2.35 
200 

.871868 
.871755 

1  .00 

1.88 

IfWl 

.952659 
.952913 

4^23 

.047341 

.047087 

7 
6 

55 

.824808 

.00 

2  OK 

.871641 

.  \f\J 

100 

.953167 

AIM 

.046833 

5 

56 

57 

.824949 
.825090 

.OO 

2.35 

i  .871528 
.871414 

.00 

1.90 

100 

.  953421 
.953675 

4i23 

.046579 
.046325 

3 

58 

.825230 

**25 

.871301 

.00 
1(\f\ 

.953929 

AIM 

.046071 

2 

59 
60 

.825371 
9.825511 

2.35 
2.33 

.871187 
9.871073 

,W 

1.90 

.954183 
9.954437 

4  '.28 

.045817 
10.045563 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.I". 

Cotang.  1  D.  1".  1  Tang. 

' 

4UO 


48° 


COSINES,  TANGENTS,  AND  COTANGENTS. 


137C 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  1".   Cotang. 

' 

0 

1 

2 

9.825511 
.825651 
.825791 

2.  as 

2.33 

9.871073 
.870960 
.870846 

1.88 
1.90 

9.954437 
.954691 
.954946 

4.23 
4  25 

10.045563 
.045309 
.045054 

60 
59 
58 

3 
4 
5 
6 

8 
9 
10 

.825931 
.826071 
.826211 
.826351 
.826491 
.826631 
.826770 
.826910 

2^33 
2.33 
2.33 
2.33 
2.33 
2.32 
2.33 
2.32 

.870732 
.870618 
.870504 
.870390 
.870276 
.870161 
.870047 
.869933 

1  .90 
1.90 
1.90 
1.90 
1.90 
1.92 
1.90 
1.90 
1.92 

.955200 
.955454 
.955708 
.955961 
.956215 
.956469 
.956723 
,956977 

4.23 
4.23 
4.23 
4.22 
4.23 
4.23 
4.23 

4.2Z 

.044800 
.044546 
.044292 
.044039 
.043785 
.04a531 
.043277 
.043023 

57 
56 
55 
54 
53 
52 
51 
50 

11 

9.827049 

9.869818 

9.957231 

10.042769 

49 

a 

.827189 
.827328 

2.33 
2.32 

.869704 
.869589 

1  .90 
1.92 

.957485 
.957739 

4.23 

.042515 
.042261 

48 
47 

14 
15 
16 
17 

.827467 
.827606 
.827745 

.827884 

2.32 
2.32 
2.32 
2.32 

.869474 
.869360 
.869245 
.869130 

1  .92 
1.90 
1.92 
1.92 

.957993 
.958247 
.958500 
.958754 

4".  23 
4.22 
4.23 

4  CM 

.042007 
.041753 
.041500 
.041246 

46 
45 
44 
43 

18 

.828023 

2.32 

.869015 

1.92 

.959008 

.&> 

A  OQ 

.040992 

42 

19 
20 

.828162 
.828301 

2.32 
2.32 
2.30 

.868900 
.868785 

1  .92 
1.92 
1.92 

.959262 
.959516 

4.><£o 

4.23 
4.22 

.0407  ,8 
.040484 

41 

40 

21 
22 

9.828439 

.828578 

2.32 

9.868670 
.868555 

1.92 

9.959769 
.960023 

10.040231 
.039977 

39 

38 

23 

.828716 

2.30 

2QO 

.868440 

1.92 

.960277 

4  22 

.039723 

37 

24 

.828855 

.  •!  * 

Son 

.868824 

1  OO 

.960530 

4  23 

.039470 

36 

25 
26 
27 

.828993 
.829131 
.829269 

.ou 

2.30 
2.30 

.868209 
.868093 
807978 

1^93 
1.92 

.960784 
.981088 
.961292 

4^23 
4.23 

.039216 
.038962 
.038708 

35 
34 

33 

28 
29 
30 

.829407 
.829545 

.829683 

2.30 
2.30 
2.30 
2.30 

.867862 
.867747 
.867631 

1.93 
1.92 
1.93 
1.93 

.961545 
I  .961799 
.962052 

4^23 
4.22 
4.23 

.038455 
.038201 
.037948 

32 
31 
30 

31 

9.829821 

9.867515 

9.962306 

4OQ 

10.037694 

29 

32 

.829959 

2'  on 

.867399 

OQ 

.962560 

.&} 

.037440 

28 

33 

.830097 

.oU 

.867283 

*'gj 

!  .962813 

4  23 

.037187 

27 

34 

.830234 

3s,  9$ 

.867167 

l  .  Jo 

!  .963067 

.036933 

20 

35 

.830372 

2.30 

.867051 

1.93 

.963320 

4OQ 

.036680 

25 

36   .830509 

2.28 

.866935 

1  .  9o 

.963574 

.309 

.036426 

24 

37   .830646 
3«   .830784 

2.28 
2.30 

.866819 
.866703 

1.93 
1.93 

.963828 
.964081 

4^22 

4  on 

.036172 
.035919 

23 
22 

39   .830921 

2.28 

.866586 

1  .95 

.964335 

.jBo 

.035665 

21 

40   .831058 

2.28 
2.28 

.866470 

1.93 
1.95 

.964588 

4!23 

.035412 

20 

41   9.831195 
42  i  .831332 

2.28 

2OQ 

9.866863 

.866237 

1.93 

9.964842 
.965095 

4.22 

A  90 

10.035158 
.034905 

19 

18 

43 
44 
45 

46 
47 

.831469 
.831606 
.831742 
.831879 
.832015 

./So 
2.28 
2.27 
2.28 

2.27 

2OQ 

.866120 
.866004 
.865887 
.865770 
.865653 

1^93 
1.95 
1.95 
1.95 

IftK 

.965349 
.965602 
.965855 
.966109 
.966362 

ft»*O 

4.22 
4.22 
4.23 
4.22^ 

.034651 
.034398 
.034145 
.033891 
.033638 

17 
16 
15 
14 
13 

48 
49 
50 

.832152 
.832288 
.832425 

./So 
2.27 
2.28 
2.27 

.865536 
.865419 
.865302 

.yo 
1.95 
1.95 
1.95 

.966616 
.966869 
.967123 

4^23 
4.22 

.033384 
.033131 
.032877 

12 
11 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9-832561 
.832697 
.832833 
.832969 
.833105 
.833241 
.833377 
.833512 
.833648 
9.8a3783 

2.27 
2.27 
2.27 
2.27 
2.27 
2.27 
2.25 
2.27 
2.25 

9.865185 
.865068 
.864950 
.  864833 
.864716 
.864598 
.864481 
.864363 
.864245 
9.864127 

1.95 

1.97 
1.95 
1.95 
1.97 
I  1.95 
!  1.97 
i  1.97 
1.97 

9.967376 
.967629 
.967883 
.968136 
.968389 
.968643 
.968896 
.969149 
.969403 
9.969656 

4.22 
4.23 
4.22 
4.22 
4.23 
4.22 
4.22 
4.23 
4.22 

10.032624 
.032371 
.032117 
.031864 
.031611 
.081357 
.031104 
.030851 
.030597 
10.030344 

9 

8 
7 
6 
5 
4 
3 
2 

0 

'  i  Cosine. 

I 

D.  1'. 

Sine.    D.  1".  1  Cotang. 

I).  1'.  i  Tang. 

' 

182' 


401 


47° 


TABLE  XXV.— LOGARITHMIC  SINES, 


138° 


1 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 

9.833783 

.833919 
.834054 

2.27 
2.25 

O  OPi 

9.864127 

.8<>4010 
.86:3892 

1.95 
1.97  ! 

9.969656 
.969909 
.970162 

4.22 
4.22 

4OQ 

10.020344 
.030091 

.029838 

60 

59 

58 

3 

.834189 

&J5     .863774 

1  .  J7  i 

.970416 

•WO 

.029584 

57 

4 

.834325 

6.6^ 

.863656 

1  .•  97  j 
1  Q7 

.970669 

4.2J 
4  fc^2 

.029331 

56 

5 
ft. 

7 

.834460 
.834595 
.834730 

2^25 
2.25 

2  OK, 

.863538 
.86:3419 
.863301 

l!98  i 
1.97  S 

.970922 
.971175 
.971429 

4.92 
4.23 

.029078 
.028825 
.028571 

55 
54 
53 

8 

.834865 

.*O 

.863183 

1  Q« 

.971682 

1  99 

.028318 

52 

9 
10 

.834999 
.835134 

2^25 
2.25 

.863064 
.862946 

l'.97  1 
1.98 

.9719&5 
.972188 

4^22 
4.22 

.028065 
.027812 

51 

50 

11 

9.835209 

0  00 

9.862827 

10*7 

9.972441 

10.027559 

49 

12 

.&35403 

.-.  .  ',') 

Q   OPv 

.862709 

.y  i 

.972695 

4.,<o 

.027305 

48 

13 
14 
15 

.835538 

.835672 
.835807 

2~23 
2.25 

.862590 
.862471 
.862353 

l!98 
1.97 

.972948 
.973201 
.973454 

4.22 
4.22 
4.22 

.027052 
.026799 
.026546 

47 
46 
45 

16 

.835941 

O  OQ 

.862234 

•  ^ 

.973707 

tf 

.026293 

44 

17 
18 

.836075 
.836209 

2^23 

Q  OQ 

.862115 
.861996 

1.'98 

.973960 
.974213 

4.22 

.026040 

.025787 

43 
42 

19 

20 

.836343 
.836477 

2^23 
2.23 

.861877 
.861758 

1  .98 
1.98 
2.00 

.974466 
.974720 

4.23 
4.22 

.025534 
.025280 

41 
40 

21 
22 
23 
24 

9.836611 
.836745 
.83687'8 
.837012 

2.23 
2.22 
2.23 

2OQ 

9.861638 
.861519 
.861400 
.861280 

1.98 
1.98 
2.00 

IOQ 

9.974973 
.975226 
.975479 
.975732 

4.22 
4.22 
4.22 

4OO 

10.025027 
.024774 
.024521 

.024268 

39 
38 
37 
36 

25 

26 

.837146 
.837279 

»j8p 
2.22 

2OO 

.861161 
.861041 

.  yo 
2.00 

.975985 
.976238 

.£& 

4.22 

.024015 

.023762 

35 
34 

27 
28 
29 
30 

.837412 
.&37S46 
.837679 
.837812 

.£& 

2.23 
2.22 
2.22 
2.22 

.860922 
.860802 
.860682 
.860562 

2^00 
2.00 
2.00 
2.00 

.976491 
.976744 
.976997 
.977250 

4^22 
4.22 
4.22 
4.22 

.023509 
.023256 
.023003 
.022750 

33 
32 
31 
30 

31 
32 
33 

9.837945 

.&38078 
.838211 

2.22 
2  22 
200 

9.860442 
.860322 
.860202 

2.00 
2.00 

9.977503 
.977756 
.978009 

4.22 
4.22 

10.022497 
.022244 
.021991 

29 

28 
27 

34 
35 

.838344 
.838477 

.*££ 
2.22 
200 

.860082 
.859962 

2.00 
2.00 

.978262 
.978515 

4.22 

4.22 

.021738 
.021485 

26 

;  25 

36 
37 

.838610 

.838742 

.£& 
2.20 

.859842 
.859721 

2.00 
2.02 

.978768 

.979021 

4.22 

4.22 

.021232 
.020979 

24 
23 

38 
39 
40 

.838875 
.839007 
.839140 

2~20 
2.22 
2.20 

.859001 
.859480 
.859360 

2.00 
2.02 
2.00 
2.02 

.97'9274 
.979527 
.979780 

4.22 
4.22 
4.22 
4.22 

.020726 
.020473 
.020220 

22 
21 

20 

41 
42 
43 
.44 

9.839272 
.839404 
.839536 
.839668 

2.20 
2.20 
2.20 

9.a5G239 
.859119 
.858998 

.858877 

2.00 

2.02 
2.02 

9.980033 

.980286 
.980538 
.980791 

4.22 
4.20 
4.22 

10.019967 
.019714 
.019462 
.019209 

19 
18 
17 
16 

45 

.839800 

2.20 

.858756 

2.02 

.981044 

4.22 

.018956 

15 

46 

.839932 

2.20 
Son 

.858635 

2.02 

.981297 

4.22 

499 

.018703 

!  14 

47 

48 

.840064 
.840196 

.!<JU 
2.20 

!  .8585-14   *•"* 
.858393   ~'°~ 

.981550 
.981803 

tJK 

4.22 

.018450 
.018197 

!  13 

1  12 

49 
50 

.840328 
.840459 

2.18 

2.20 

i  .858272 
!  .858151 

«.u« 
2.02 
2.03 

.982056 
.982309 

4.  '22 

4.22 

.017944 
.017691 

!  11 
10 

51 

52 

9.840591 
.840722 

0  18    9.858029   0  00 

/£  .  1  0        ft^'*'Qftft      * 

9.982562 
.982814   T'S! 

10.017438 
.017186 

9 
8 

53 
54 

.840854 
.840985 

2.20 

2.18 

O  1O 

«.Vr'8R    'a-uo 
.00«ttO    o  flg 

.857665   *.02 

.983067 
.983:320 

4.22 

<  k).i 

.01G933 
.016680 

7 
6 

55 

.841116 

6.  10 

21  Q 

.857543   ~J5    .983573   *•£ 

.016427 

5 

56 

.841247 

.  Jo 

2-tQ 

.857422   rXo   i  .983826 

V^WV 

A   .».» 

.01617'4 

4 

57 

.841378 

.10 

.857300   5-2 

.984079   r2S 

.015921 

3 

58 
59 
60 

.841509 
.841640 
9.841771 

2.18 

2.18 
2.18 

.857178 
.857056 
9.856934 

JS.UO 

2.03 
2.03 

.984,332 
.984584 
9.984837 

4.20 
4.22 

.015668 
.015416 
10.015163 

2 
1 
0 

' 

Cosine. 

D.I". 

Sine,  i  D.  1". 

Cotang.  D.  1".    Tang.   ' 

133= 


403 


COSINES,  TANGENTS,  AND  COTANGENTS. 


135= 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

9.841771 
.841902 

2.18 

9.856934 

.856812 

2.03 

9.984837 
.985090 

4.22 

10.015163 
.014910 

60 
59 

i) 
3 
4 
5 
6 

rt 

.842033 
.842163 
.842294 
.842424 
.842555 
.8426a5 

2.  18 
2.17 
2.18 
2.17 
2.18 
2.17 

O  1*7 

.856690 
.856568 
.856446 
.856323 
.856201 
.856078 

2^03 
2.03 
2.05 
2.03 
2.05 

.985343 
.985596 
.985848 
.986101 
.986354 
.986607 

4.22 
4.22 
4.20 
4.22 
4.22 
4.22 

.014657 
.014404 
.014152 
.013899 
.013646 
.013393 

58 
57 
56 
55 
54 
53 

8 
9 
10 

.842815 
.842946 
.843076 

ir.Jf 

2.18 
2.17 
2.17 

.855956 
.855833 
.855711 

2!05 
2.03 
2.05 

.986860 
.987112 
.987365 

4^20 
4.22 
4.22 

.013140 
.012888 
.012635 

52 
51 
50 

11 
12 

9.843206 
.843336 

2.17 
2  1  "^ 

9.855588 
.855465 

2.05 

9.987618 

.987871 

4.22 

10.012382 
.012129 

49 

48 

13 
14 
15 

16 

.843466 
.843595 
.843725 
.843855 

2^15 
2.17 
2.17 

.855342 
.855219 
.855096 
.854973 

2.05 
2.05 
2.05 
2.05 

.988123 
.988376 
.988625) 

.988882 

4.20 
4.22 
4.22 
4.22 

.011877 
.011624 
.011371 
.011118 

47 
46 
45 
44 

17 

.843984 

2.15 

21  "7 

.854850 

2.05 

.989134 

4.20 

.010866 

43 

18 
19 

.844114 
.844243 

.  1  1 

2.15 

21  PC 

.854727 
.854603 

2.05 

2.07 

.989387 
.989640 

4.22 
4.22 

.010613 
.010360 

42 
41 

20 

.844372 

.  lo 

2.17 

.854480 

2.05 
2.07 

.989893 

4^20 

.010107 

40 

21 
22 

9.814502 
.814631 

2.15 

9.854356 
.854233 

2.05 

9.990145 
.990398 

4.22 

10.009855 
.009602 

39 

38 

23 

24 
25 

26 
27 
23 
29 
30 

.844760 
.844889 
.845018 
.845147 
.845276 
.845405 
845533 
.845662 

2.  15 
2.15 
2.15 
2.15  . 
2.15 
2.15 
2.13 
2.15 
2.13 

.854103 
.853986 
.853862 
.853738 
.853614 
.853490 
.853366 
.853242 

2.07 
2.05 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 

.990651 
.990903 
.991156 
.991409 
.991662 
.991914 
.992167 
.992420 

4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 

.009349 
.009097 
.008844 
.008591 
.008338 
.008086 
.007833 
.007580 

37 
36 
35 
34 
33 
32 
31 
30 

31 
32 

9.845790 
.845919 

2.15 

21  Q 

9.853118 

.852994 

2.07 

2AQ 

9.992672 
.992925 

4.22 

10.007328 
.00707'5 

29 

28 

33 
34 
35 
36 
37 
38 
39 
40 

.840047 
.846175 
.846304 
.846432 
.846560 
.846688 
.816816 
.846944 

.  lo 
2.13 
2.15  i 
2.13 
2.13 
2.13  i 
2.13 
2.13 
2.12 

.852869 
.852745 
.852620 
.852496 
.852371 
.852247 
.852122 
.851997 

.Uo 
2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.08 
2.08 

.993178 
.993431 
.993683 
.993936 
.994189 
.994441 
.994694 
.994947 

4.22 
4.22 
4.20 
4  22 
4^22 
4.20 
4.22 
4.22 
4.20 

.006822 
.006569 
.006317 
.006064 
.005811 
.005559 
.005306 
.005053 

27 
26 
25 
24 
23 
22 
21 
20 

41 
42 

9.847071 
.847199 

2.13 

210 

9.851872 
.851747 

2.08 

2AQ 

9.995199 
.995452 

4.22 
400 

10.004801 
.004548 

19 

18 

43 

.847327 

.  lo 

.851622 

.Uo 

.995705 

.£& 

.004295 

17 

44 
45 

46 
47 
48 

.847454 
.847582 
.847709 
.847836 
.847964 

2.  12 
2.13 
2.12 
2.12 
2.13 

.851497 
.851372 
.851246 
.851121 
.850996 

2.08 
2.08 
2.10 
2.08 
2.08 

.995957 
.996210 
.996463 
.996715 
.996968 

4.20 
4.22 
4.22 

4.20  - 
4.22 

.004043 
.003790 
.003537 
.003285 
.003032 

16 
15 
14 
13 
12 

49 
50 

.848091 
.848218 

2.12 
2.12 
2.12 

.850870 
.850745 

2.10 
2.08 
2.10 

.997221 
.997473 

4.22 
4.20 
4.22 

.002779 

.002527 

11 
10 

51 
52 
53 
54 
55 
56 
57 

9.848345 

.848472 
.848599 
.848726 
.848a52 
.848979 
.849106 

2.12 
2.12 
2.12 
2.10 
2.12 
2.12 

21  A 

9.850619 
.850493 
.850368 
.850242 
.850116 
.849990 
.849864 

2.10 
2.08 
2.10 
2.10 
2.10 
2.10 

9.997726 
.997979 
.998231 
.998484 
'  .998737 
.998989 
.999242 

4.22 
4.20 
4.22 
4.22 
4.20 
4.22 

10.002274 
.002021 
.001769 
.001516 
.001263 
.001011 
.000758 

9 
8 
7 
6 
5 
4 
3 

58 
59 
60 

.849232 
.849359 
9.849485 

.1U 
2.12 
2.10 

.849738 
.849611 
9.849485 

2.10 
2.12 
2.10 

.999495 
.999747 
0.000000 

4.22 
4.20 
4.22 

.000505 
.000253 
10.000000 

2 
1 

0 

' 

Cosine. 

"^77 

Sine. 

D.  1". 

Cotang. 

D.  1'. 

Tang. 

' 

45° 


4C3 


TABLE  XXVI.-LOGARITHMIC  YERSED  SIXES. 


TABLE  XXVI.-LOGARITHMIC  VERSED  SINES 


0° 

1° 

It 

' 

Vers. 

q-2l 

Ex.  sec. 

// 

f 

Vers. 

q-2l 

Ex.  sec. 

9.070 

9.07'0 

0 

0  1  Inf.  neg.!  120 

120  Inf.  neg. 

3600 

0  6.182714 

109  !  175 

6.182780 

60 

1  2.626422  120; 

120  2.626422 

3660 

1 

.197071 

108  177 

.197139 

120 

2  3.228482  120 

120  .3.228482 

3720 

2 

.211194 

108  179 

.211264 

180 

3  i  .580665  i  120  i 

120  |  .580665 

3780 

3 

.225091 

108:  1  181 

.225164 

340 

4  3.830542  i  120  i 

120  3.830542 

3840 

4 

.238770 

107  i  182 

.238845 

300 

5  4.024362 

120 

120  4.024363 

3900 

5 

.252236 

107  184 

.252314 

3(50 

6 

.182725 

120 

120 

.182725 

3960 

6 

.265497 

106  !  186 

.265577 

420 

7 

.316618 

120 

120 

.316619 

4020 

7 

.278558 

106 

188 

.278641 

480 

8 

.432602 

120 

121 

.432603 

4080 

8 

.291426 

106 

!  191 

.291511 

540 

9 

.534907 

119 

121 

.534908 

4140 

9 

.304106 

105 

193 

.304193 

600 

10 

.626422 

119 

121 

.626424 

4200 

10 

.316603 

105 

|195 

.316693 

660 

11 

4.709207 

119 

122 

4.709209 

4260 

11 

6.328923 

104 

197 

6.329016 

720 

12 

.784784 

119 

122 

.784787 

4320 

12 

.341071 

104  jf  199 

.341167 

780 

13 

.854308 

119 

122 

.854312 

4380 

13 

.353052 

108  201 

.353150 

840 

14 

.918678 

119 

123 

.918681 

4440 

14 

.364869 

103  204 

.364970 

900 

15 

4.978604 

119 

123 

4.978608 

4500 

15 

.376528 

103  j  206 

.376631 

960 

16 

5.034661 

119 

124  5.034666 

4560 

16 

.388032. 

102  208 

.388138 

1020 

17 

.087319 

119 

124 

.087325 

4620 

17 

.399386 

102  211 

.399494 

1080 

18 

.136966 

119 

125 

.136972 

4680 

18 

.410593 

101  213 

.410705 

1140 

19 

.183928 

119 

125 

.183935 

4740 

19 

.421657 

101 

215 

.421772 

1200 

20 

.228481 

119 

126 

.228488 

4800 

20 

.432583 

100 

,218 

.4327'00 

1260 

21 

5.270859 

118 

126 

5.270868 

4860 

21 

6.443372 

100 

!220 

6.443493 

1320 

22 

.311266 

118 

127 

.311275 

4920 

22 

.454029 

099  !223 

.454153 

1380 

23 

.349876 

118 

128 

.349886 

4980 

23 

.464557 

099 

225 

.464684 

1440 

24 

.386843 

118 

129 

.386854 

5040 

24 

.474959 

098 

±2S 

.475089 

1500 

25 

.422300 

118 

129 

.422312 

5100 

25 

.4a5238 

098  ;  :  230 

.485371 

1560 

26 

.456367 

118 

130 

.456379 

5160 

26 

.495396 

097  233 

.495532 

1620 

27 

.489148 

118 

131 

.489161 

5220 

27 

.505438 

097  236 

.505577 

1680 

28 

.520736 

117 

132 

.520750 

5280 

28 

.515364 

09(3  238 

.515506 

1740 

29 

.551216 

117 

133 

.551231 

5340 

29 

.525178 

095  i  241 

.525324 

1800 

30 

.580662 

117 

134 

.580679 

5400 

30 

.534882 

095 

MM4 

.535031 

1860 

31 

5.609143 

117 

134 

5.609160 

5460 

31 

6.544480 

094 

'247 

6.544632 

1920 

32 

.636719 

117 

135 

.636738 

5520 

32 

.553972 

094 

249 

.554128 

1980 

33 

.663447 

116 

136 

.663467 

5580 

33 

.56*362  093 

252 

.563521 

2040 

34 

.689377 

116 

137 

.6891398 

5640 

34 

.572651  (,93 

255 

.572813 

2100 

&5 

.714555 

116 

138 

.714577 

5700 

35 

.581842  !092 

258 

.582008 

2160 

36 

.739033 

116 

140 

..739047 

5760 

36 

.590936 

092 

261 

.591106 

2220 

37 

.762821 

116 

141 

.762847 

5820 

37 

.599937 

091 

564 

.600110 

2280 

38 

.785985 

115 

142 

.786012 

5880 

38 

.608845 

090  267 

.609021 

2340 

39 

.808547 

115 

143 

.808575 

5940 

39 

.617663 

090  i  27'0 

.617843 

2400 

40 

.830537 

115 

144 

.830567 

6000 

40 

.626392 

089 

273 

.626575 

2460 

41 

5.R51985 

115 

145 

5.852016 

6060 

41 

6.635034 

089 

!276 

6.635221 

2520 

42 

.872915 

114 

147 

.872948  1  6120 

42 

.643591 

088  279 

.643782 

2580 

43 

.893353 

114 

148 

.893:387   6180 

43 

.652064 

087  1  282 

.652259 

2640 

44 

.913:322 

114 

149 

.913357 

6240 

44 

.660456 

087  1  285 

.660655 

2700 

45 

.932841 

114 

151 

.932878 

6300 

45 

.668767 

086!  289 

.668970 

2760 

46 

.951931 

113 

152 

.951970 

6360 

46 

.677000 

085  292 

.677206 

2820 

47 

.970611 

113 

154 

.970652 

6420 

47 

.685155 

085 

295 

.685365 

2880 

48 

5.988898 

113 

155  5.988940 

6480 

48 

.693234 

084 

298   .69:3448 

2940 

49 

6.006807 

112 

157  i  6.  00(5851 

6540 

49 

.701239 

083 

302   .701457 

3000 

50 

.024355 

112 

158 

.024401 

6600 

50 

.709171 

083 

305 

.709393 

3060 

51 

6.041555 

112 

160 

6.041602 

6660 

51 

6.717030 

082 

308 

6.717257 

3120 

52 

.058421  111 

161 

.058470 

6720 

52 

.724820 

081 

312 

.725050 

3180 

53 

.074965  111 

163 

.075017 

6780 

53 

.732540 

081 

315 

.732775 

3240 

54 

.091201 

111 

164 

.091254 

68-10 

54 

.740192 

080 

319 

.740431 

3300 

55 

.107138 

110 

166 

.107194 

6900 

55 

.747777 

079 

322 

.748020 

3360 

56 

.122789 

110 

168 

.122846 

6960 

56 

.755297 

079 

326 

.755544 

3420 

57 

.138162 

110 

169 

.138222 

7020^ 

57 

.762752 

078 

329 

.763004 

3480 

58 

.153268 

109 

171 

.15*330 

7080 

58 

.770144 

077 

333 

.770400 

3540 

59 

.168116 

109 

173 

.168180 

7140 

59 

.777473 

076 

337 

.777733 

3600 

60 

6.182714 

109 

175 

6.182780 

7200 

60 

6.784741 

076 

340 

6.785005 

a 

/ 

ft 

/ 

404 


AND  EXTERNAL  SECANTS. 


2° 

3° 

"  i' 

Vers. 

q-2l 

Ex.  sec. 

" 

' 

Vers. 

q-  21 

Ex.  sec. 

M 

9.070 

9.070* 

7200!  0  6.784741 

076  !  340  '6.  785005 

10800 

017.136868 

021 

616 

7.137464 

7260  1 

.791948 

075 

344 

.792217 

10860 

1 

.141679 

Ot9 

622 

.142281 

7320  2 

.799096 

074 

348 

.799370 

10920 

g 

.146464 

018 

627 

.147072 

7380  3 

.806186 

073 

351 

.8064&4 

10980 

8 

.151222 

017 

633 

.151837 

7440  4 

.813219 

073 

355 

.813501 

11040 

4 

.155954 

016 

638 

.156577 

7500  5 

.820194 

072 

359 

.820482 

11100 

5 

.160661 

015 

644 

.161290 

7560  6 

.827115 

071 

303 

.827406 

11160 

6 

.165342 

014 

650 

.165978 

7620  7 

.833980 

070 

3(J7 

.831277 

11220 

7 

.169998 

013 

655 

.170641 

7680  8 

.840792 

070 

371 

.841093 

11280 

8 

.174630 

Oil 

661 

.175279 

7740!  9 

.847551 

069 

375 

.847857 

11340 

9 

.179236 

010 

C67 

.179893 

7800  10 

.854257. 

068 

379 

.854568 

11400 

10 

.18:3819 

009 

673 

.184483 

7860  '  11 

6.860912 

067 

383 

6.861228 

11460 

11 

7.188377 

008 

679 

7.189048 

7920  12 

.867517 

066 

387 

.867837 

11520 

12  .192912 

007 

685 

.193589 

7980  13 

.874071 

066 

391 

.874396 

11580 

13  .197423 

006 

690 

.198108 

8040  14 

.880577 

065 

395 

.880907 

11640 

14 

.201910 

004 

696 

.202602 

8100  15 

.8870:34 

064 

399 

.887369 

11700 

15 

.206375 

003 

702 

.207074 

8160  16 

.893443 

063 

403 

.893783 

11760 

it; 

.210817 

002 

708 

,211523 

8220  17 

.899806 

062 

407 

.900151 

11820 

17 

.215236 

001 

714 

.215949 

8280  18 

.906122 

061 

411 

.906472 

11880 

is 

219633 

400 

720 

.220353 

8340  19 

.912393 

061 

416 

.912748 

11940 

1!) 

.224007 

998 

727 

.224735 

8400  20 

.918618 

060 

420 

.918979 

12000 

20 

.228360 

997 

733 

.229095 

8460  21 

6.924800 

059 

424 

6.925165 

12060 

21 

7.232691 

996 

739 

7.233433 

8520  22 

.930937 

058 

429 

.931308 

12120 

22 

.237000 

995 

745 

.237750 

8580  33 

.937032 

057 

433 

.937408 

12180 

23 

.241288 

994 

751 

.242046 

8640J24 

.943084 

056 

437 

.943465 

12240 

24 

.245555 

992 

757 

.246320 

8700^25 

.949094 

055 

442 

.949480 

12300 

25 

.249801 

991  i 

764 

.250574 

8760  26 

.955063 

054 

446 

.955455 

12360 

26 

.254027 

990 

770 

.254807 

8820  27 

.960991 

054 

451 

.961388 

12420 

27 

.258232 

989  ' 

776 

.259019 

8880J28 

.966879 

053 

455 

.967281 

12480 

28 

.262416 

987 

783 

.263212 

8940  29 

.972727 

052 

460 

.973135 

12540 

29 

.266581 

986 

789 

.267384 

9000J30 

.978536 

051 

464 

.978949 

12600 

30 

.270726 

985 

795 

.271537 

9060  '31 

6.984306 

050 

469 

6.984725 

12660 

81 

7.274851 

9&3 

802 

7.275669 

9120  32 

.990039 

049 

i474 

.990463 

12720 

32 

.278956 

982 

808 

.279783 

9180  33 

6.995733 

048  :  478 

6.996164 

13780 

.283043 

981 

815 

.283877 

9240  !  34 

7.001391 

047 

483 

7.001827 

12840 

84 

.287110 

980 

821 

.287952 

9300  35 

.007012 

046 

488 

.007454 

12900 

35 

.291158 

97'8 

828 

.292007 

9360  36 

.012597 

045 

493 

.013044 

12960 

.295187 

977 

835 

.296045 

9420  37 

.018146 

044 

497 

.018599 

13020 

87 

.299197 

976 

841 

.300063 

9480  38 

.023660 

043 

502 

.024119 

13080 

3S 

.303190 

974 

848 

.304063 

9540  39 

.029139 

042 

507 

.029604 

13140 

39 

.307164 

973 

855 

.308045 

9600j40 

.034584 

041 

512 

.035054 

13200 

40 

.311119 

972 

861 

.312009 

9660141 

7.039995 

040 

517 

7.040471 

13260 

41 

7.315057 

970 

868 

7.315955 

9720  i  42 

.045372 

039 

522 

.045854 

13320 

42 

.318977 

969 

875 

.319883 

9780  43 

.050716 

038 

527 

.051204 

13380 

43 

.322880 

967 

882 

.323794 

9840  44 

.056028 

037 

532 

.056522 

13440 

44 

.326765 

966 

889 

.327687 

9900  45 

.061:307 

036 

587 

.061807 

13500 

45 

.330632 

968 

896 

.331563 

9960  !  46 

.066554 

035 

542 

.067061 

13560 

46 

.334483 

963 

902 

.335422 

10020  [47 

.071770 

034 

547 

.072282 

13620 

47 

.338316 

962 

908 

.339263 

10080  48 

.076954 

033 

552 

.077473 

13680 

48 

.342133 

961 

916 

.343089 

10140  49 

.082108 

032 

557 

.082633 

13740 

4!) 

.3459a3 

959 

923 

.346897 

10200  50 

.087232 

031 

562 

.087763 

13800 

no 

.349716 

958 

930 

.350689 

10260  51 

7.092325 

030 

568 

7.092862 

13860 

51 

7.353483 

956 

938 

7.354464 

10320  52  i  .097:389 

029 

5731  .097932 

13920 

52 

.357233 

955 

945 

.358223 

10380  53 

.102423 

028 

578 

.102973 

13980 

58 

.360968 

953 

952 

.361966 

10440!  54 

.107428 

027 

584 

.107985 

14040 

54 

.364686 

952 

959 

.365693 

lO'iOO  55 

.112405 

026 

589 

.112968 

14100 

55 

.368389 

951 

966 

.369404 

10560  56 

.117353 

025 

594 

.117922 

141  GO 

56 

.372076 

949 

973 

.373100 

10620  57 

.122273 

024 

600 

.122849 

14220 

57 

.375747 

948 

981 

.376780 

10680  58 

.127165 

023 

1605 

.127748 

14280 

58 

.3794a3 

946 

988 

.380444 

10740  59 

.1320:30 

022 

611 

.132619 

14340 

59 

.383043 

945 

995 

.384094 

10800  60 

7.136868 

021 

616 

7.137464 

14400 

60 

7.386668 

943 

403 

7.387728 

«   » 

9.069*1  9.071* 

405 


TABLE  XXVI.-LOGARITHMIO  VERSED  SINES 


4° 

5° 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  r. 

' 

Vers. 

D.I'. 

Ex.  sec.  D.  1*. 

0 

7.386668  60.17  7.387728 

60.32 

0 

7.580389 

48.15  7.582045  i  48.33 

1 

.390278 

59.93   .391347 

60.07 

1 

.883278 

47.98   .584945  48.17 

2 

.393874 

59.67   .394951 

59.82 

2 

.586157 

47.82   .587835  |  48.00 

3 

.397454  59.43 

.398540 

59.57 

3 

.589026 

47.67 

.590715  !  47.87 

4 

.401020 

59.18 

.402114 

59.33 

4 

.591886 

47.52 

.593587  i  47.70 

5 

.404571 

58.93 

.405674 

59.10 

5 

.504737 

47.35 

.596449 

47.53 

6 

.408107 

58.70 

.409220 

58.85 

6 

.597578 

47.20 

.599301 

47.38 

7 

.411629 

58.47 

.412751 

58.62 

7 

.600410 

47.05 

.602144  47.25 

8 

.415137 

58.23 

.416268 

58.38 

8 

.603233 

46.90 

.604979 

47.08 

9 

.418631 

58.00 

.419771 

58.15 

9 

.606047 

47.73 

.607804 

46.92 

10 

.422111 

57.77 

.423260 

57.92 

10 

.608851 

46.00 

.610619 

46.78 

11 

7.425577 

67.58 

7.426735 

57.70 

11 

7.611647 

46.43 

7.613426 

46.63 

12 

.429029 

57.30 

.430197 

57.45 

12 

.614433 

46.30 

.616224 

46.48 

13  1  .432467 

57.08 

.433644 

57.25 

13 

.617211 

46.15 

.619013 

46.35 

14 

.435892 

56.85 

.437079 

57.00 

14 

.619980 

45.98 

.621794 

46.18 

15 

.439303 

56.63 

.440499 

56.80 

15 

.622739 

45.87 

.624565 

46.05 

16 

.442701 

56.42 

.443907  56.57 

16 

.625491 

45.70 

.627328 

45.90 

17 

.446086 

56.20 

.447301 

56.35 

17 

.628233 

45.57 

.630082  45.75 

18 

.449458 

55.97 

.450682 

56.13 

18 

.630967 

45.42 

.632827 

45.62 

19 

.452816 

55.77 

.454050 

55.92 

19 

.633692 

45.28 

.635564 

45.48 

20 

.456162  55.55 

.457405 

55.72 

20 

.636409 

45.13 

.638293 

45.33 

21 

7.459495 

55.  as 

7.460748 

55.48 

21 

7.639117 

44.98 

7.641013 

45.18 

22 

.462815 

55.12 

.464077 

55.28 

22 

.641816 

44.87 

.643724 

45.07 

23 

.466122  54.92 

.467394 

55.08 

23 

.644508 

44.72 

.646428 

44.90 

24 

.469417 

54.70 

.470699 

54.87 

24 

.647191 

44.57 

.649122 

44.78 

25 

.  .472699 

54.50 

.473991 

54.65 

25 

.649865 

44.45 

.651809 

44.65 

26 

.475969 

54.28 

.477270 

54.47 

26 

.652532 

44.30 

.654488 

44.50 

27 

.479226 

54.10 

.480538 

54.25 

27 

.655190 

44.17 

.657158 

44.37 

28 

.482472 

53.88 

.483793 

54.05 

28 

.657840 

44.05 

.659820 

44.23 

29 

.485705 

53.70 

.487036 

53.85 

29 

.660483 

43.90 

.662474 

44.12 

30 

.488927 

53.48 

.490267 

53.67 

30 

.663117 

43.77 

.665121 

43.97 

31 

7.492136 

53.28 

7.493487 

53.45 

31 

7.665743 

43.63 

7.667759 

43.  83 

32 

.495333 

53.10 

.496694 

53.27 

32 

.668361 

43.50 

.670389 

43.73 

33 

.498519 

52.90 

.499890 

53.07 

33 

.670971 

43.38 

.673012 

43.57 

34 

.501693 

52.72 

.503074 

52.88 

34 

.673574 

43.23 

.675626 

43.45 

35 

.504856 

52.52 

.506247 

52.68 

35 

.676168 

43.12 

.678233 

43.33 

36 

.508007 

52.33 

.509408 

52.50 

36 

.678755 

42.98 

.680833 

43.18 

37 

.511147 

52.13 

.512558 

52.32 

37 

.681334 

42.87 

.683424 

43.07 

38 

.514275 

51.95 

.515697 

52.12 

38 

.683906 

42.73 

.686008 

42.95 

39 

.517392 

51.77 

.518824 

51.93 

39 

.686470 

42.60 

.688585 

42.82 

40 

.520498 

51.58 

.521940 

51.77 

40 

.689026 

42.48 

.691154 

42.68 

41 

7.523593 

51.40 

7.525046 

51.57 

41 

7.691575 

42.35 

7.693715 

42.57 

42 

.526677 

51.22 

.528140 

51.38 

42 

.694116 

42.  x3 

.690269 

42.43 

43 

.529750 

51.03 

.531223 

51.22 

43 

.696650 

42.12 

.698815 

42.33 

44 

.532812 

50.85 

.534296 

51.02 

44 

.699177 

41.98 

.701355 

42.20 

45 

.535863 

50.68 

.537357 

50.85 

45 

.701696 

41.87 

.703887 

42.07 

46 

.538904 

50.50 

.540408 

50.68 

46 

.7'0420B 

41.73 

.706411   41.97 

47 

.541934 

50.32 

.543449 

50.50 

47 

.706712 

41.63 

.708029  !  41.83 

48 

.544953 

50.15 

.546479 

50.33 

48 

.709210 

41.50 

.711439 

41.72 

49 

.547962 

49.98 

.549499 

50.15 

49 

.711700 

41.88 

.713942 

41.60 

50 

.550961 

49.80 

.552508 

49.98 

50 

.714183 

41.27 

.716438 

41.48 

51 

7.553949 

49.63 

7.555507 

49.80 

51 

7.716659 

41.15 

7.718927 

41.37 

52 

.556927 

49.43 

.558495 

49.65 

52 

.719128 

41.03 

.721409 

41.25 

68 

.559395 

49.28 

.561474 

49.47 

53 

.721590  !  40.92 

.723884 

41.13 

54 

.562852 

49.13 

.564442 

49.32 

54 

.724045 

40.80 

.726352 

41.02 

55 

.565800 

48.95 

.567401 

49.13 

55 

.726493 

40.68 

.728813 

40.90 

56 

.568737 

48.80 

.570349 

48.98 

56 

.728934 

40.57 

.731267 

40.78 

57 

.571665 

48.63 

.573288 

48.82 

57 

.731368 

40.47 

.733714 

40.68 

53 

.574583 

48.47 

.576217 

48.65 

58 

.733796 

40.33 

!736155 

40.57 

59 

.577491 

48.30 

.579136 

48.48 

59 

.736216 

40.23 

.738589 

40.45 

GO 

7.580389 

48.15 

7.582045 

48.33 

60 

7.738630 

40.13  7.741016 

40.33 

406 


AND  EXTERNAL  SECANTS. 


6° 

7° 

' 

Vers. 

D.  1". 

Ex.  sec. 

D.I". 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  1'. 

9 

7.738630  i  40.13 

7.741016  !  40.33    0 

7.872381 

34.38  7.875630 

34.63 

1 

.741038 

40.00 

.743436  40.23  1   1 

.874444 

34.30 

.877708 

34.57 

2 

.743438 

39.90 

.745850  40.13  !  2 

.876502 

34.22 

.879782 

34.48 

3 

.745832 

39.78 

.748258 

40.00  !  3 

.878555 

34.13 

.881851 

34.40 

4 

.748219 

89.68 

.750658 

39.90    4 

.880603 

34.07 

.883915 

34.32 

5 

.750600 

39.57 

.753052 

39.80 

5 

.882647 

33.98 

.885974 

34.25 

6 

.752974 

39.47 

.755440  39.68 

6 

.884686 

33.90 

.888029 

34.15 

°7 

.755342 

39.  a5 

.757821 

39.58 

7 

.886720 

33.82 

.890078 

34.08 

8 

.757703 

39.25 

.760196 

39.48 

8 

.888749 

33.73 

.892123 

34.02 

9 

.760058 

39.13 

.762565 

39.37 

9 

.890773 

as.  67 

.894164 

33.92 

10   .762406 

39.05 

.764927 

39.25 

10 

.892793 

33.58 

.896199 

33.85 

11   7.764749 

38.92 

7.767282 

39.17 

11 

7.894808 

33.50 

7.898230 

33.77 

12  \  .767084 

38.83 

.769632 

39.05 

12 

.896818 

33.43 

.900256 

33.70 

13 

.769414 

38.72 

.771975 

38.95 

13 

.898824 

33.35 

.902278 

33.62 

14 

.771737 

38.62 

.774312 

38.85 

14 

.900825 

33.27 

.904295 

33.53 

15 

.774054 

38.52 

.776643 

38.75 

15 

.902821 

33.20 

.906307 

33.47 

16 

.776365 

38.42 

.778968 

38.63 

16 

.904813 

33.12 

.908315 

33.40 

17 

.778670 

38.30 

.781286 

38.55 

17 

.906800 

33.05 

.910319 

as.  so 

18 

.780968 

38.22 

.783599 

38.43 

18 

.908783 

32.  9/ 

.912317 

33.25 

19 

.783-261 

38.10 

.785905 

38.35 

19 

.910761 

32.90 

.914312 

33.17 

20 

.785547 

38.02 

.788206 

38.23 

20 

.912735 

32.82 

.916302 

33.08 

21 

7.787828 

37.90 

7.790500 

38.15 

21 

7.914704 

32.73 

7.918287 

33.02 

22 

.790102 

37.82 

.792789 

38.03 

22 

.916668 

32.68 

.920268 

32.95 

23 

.792371 

37.70 

.795071 

37.95 

23 

.918629 

32.58 

.922245 

32.87 

24 

.794633 

37.62 

.797348 

37.85 

24 

.920584 

32.53 

.924217 

32.78 

23 

.796890 

37.52 

.799619 

37.75 

25 

.922536 

32.45 

.926184 

•32.73 

26 

.799141 

37.40 

.801884 

37.65 

26 

.924483 

32.37 

.928148 

32.65 

27 

.801:385 

37.  as 

.804143 

37.57 

27 

.926425 

32.32 

.930107 

32.58 

28 

.803625 

37.22 

.806397 

37.45  • 

28 

-.928364 

32.22 

.932062 

32.50 

29 

.805858 

37.13 

.808644 

37.37 

29 

.930297 

32.17 

.934012 

32.43 

30 

.808086 

37.03 

.810886 

37.28 

30 

.932227 

32.08 

.935958 

32.37 

31 

7.810308 

36.93 

7.813123 

37.17 

31 

7.934152 

32.02 

7.937900 

32.30 

32 

.812524 

36.83 

.815353 

37.08 

32 

.936073 

31.95 

.939838 

32.23 

33 

.814734 

36.75 

.817578 

37.00 

33 

.937990 

81.88 

.941772 

32.15 

34 

.816939 

36.67 

.819798 

36.90 

34 

.939903 

31.80 

.943701 

32.08 

35 

.819139 

36.55 

.822012 

36.80, 

35 

.941811 

31.73 

.945626 

32.02 

36 

.821332 

36.48 

.824220 

36.72 

36 

.943715 

31.67 

.947517 

31.95 

37 

.823521 

36.37 

.826423 

36.62 

37 

.945615 

31.60 

.949464 

31.87 

38 

.825703 

36.28 

.828620 

36.53 

38. 

.947511 

31.52 

.951376 

31.82 

39 

.827880 

36.20 

.830812 

36.45 

39 

.949402 

31.47 

.953285 

31.73 

40 

.830052 

36.10 

.832999 

36.35 

40 

.951290 

31.38 

.955189 

31.68 

41 

7.832218 

36.03 

7.&35180 

36.27 

41 

7.953173 

31.32 

7.957090 

31.60 

42 

.834379 

35.93 

.837356 

36.17 

42 

.955052 

31.27 

.958986 

31.53 

43 

.836535 

35.83 

.839526 

36.08 

43 

.956928 

31.18 

.960878 

31.48 

44 

.838685 

35.75 

.841691 

36.00 

44 

.958799 

31.12 

.962767 

31.40 

45 

.840830 

35.65 

.843851 

35.90 

45 

.960666 

31.05 

.964651 

31.35 

46 

.842969 

35.58 

.846005 

35.83 

46 

.962529 

30.98 

.966531 

31.28 

47 

.845104 

35.48 

.848155 

35.73 

47 

.964388 

30.92 

.968408 

31.20 

48 

.847233 

85.40 

.850299 

35.63 

48 

.966243 

30.85 

.970280 

31.13 

49 

.849357 

85.30 

.852437 

85.67 

49 

.968094 

30.78 

.972148 

31.08 

50 

.851475 

35.23 

.854571 

35.48 

50 

.969941 

30.73 

.974013 

31.02 

51 

7.853589 

85.18 

7.868700 

35.38 

51 

7.971785 

30.65 

7.975874 

30.93 

52 

.855697 

35.05 

.858823 

35.32  j  52 

.973624 

30.58 

.977730 

30.88 

53 

.857800 

34.97 

.860942 

35.22  I!  53 

.975459 

30.53 

.979583 

30.82 

54 

.859898 

34.88 

.863055 

35.13 

54 

.977291 

80.45 

.981432 

30.75 

55 

.861991 

34.80 

.865163 

35.05 

55 

.979118 

30.40 

.983277 

30.70 

56 

.864079 

34.72 

.867266 

34.98 

56 

.980942 

30.33 

.985119 

30.62 

57 

.866162 

34.63 

.869365 

34.88 

57 

.982762 

80.27 

.986956 

30.57 

58 

.868240 

34.55 

.871458 

34.80 

58 

.984578 

30.22 

.988790 

30.50 

59 

.870313 

34.47 

.873546 

34.73 

K(\ 

.986391 

30.13 

.990620 

30.43 

GO 

7.  872381 

34.38 

7.875630 

34.63 

CO 

7.988199 

30.08 

7.992446 

30.38 

407 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


8° 

9° 

/ 

Ver». 

D.  1". 

Ex.  sec. 

D.  1". 

f 

Vers. 

D.  1" 

Ex.  sec 

D  1" 

0 

7.988199 

30.08 

7.992446 

30.38  ! 

0 

8.090317 

26.72 

8.095697 

27.05 

1 

.990004 

30.02 

.994269 

30.32 

1 

.091920 

26.68 

.097320 

27.02 

2 

.991805 

29.95 

.996088 

30.25 

2 

.093521 

26.63 

.098941 

26.97 

3 

.993602 

29.88 

.997903 

30.18 

3 

.095119 

26.58 

.100559 

26.92 

4 

.995395 

29.83 

7.999714 

30.13 

4 

.096714 

26.52 

.102174 

26.87 

5 

.997185 

29.77 

8.001522 

30.07 

5 

.098305 

26.48 

.103786 

26.82 

6 

7.998971 

29.72 

.003326 

30.00 

6 

.099894 

26.43 

.105395 

26.77 

7 

8.000754 

29.63 

.005126 

29.95 

7 

.101480 

26.40 

.107001 

26.73 

8 

.002532 

29.60 

.006923 

29.88 

8 

.103064 

26.33 

.108605 

26.67 

9 

.004308 

29.52 

.008716 

29.83 

9 

.104644 

26.28 

.110205 

26.63 

10 

.006079 

29.47 

.010506 

29.73 

10 

.106221 

26.25 

.111803 

26.58 

11 

8.007847 

29.40 

8.012292 

29.70 

11 

8.107796 

26.18 

8.113398 

26.53 

12 

.009611 

29.35 

.014074 

29.65 

12 

.109367 

26.15 

.114990 

26.48 

13 

.011372 

29.28 

.015853 

29.58 

13 

.110936 

26.10 

.116579 

26.45 

14 

.013129 

29.22 

.017628 

29.53 

14 

.112502 

26.05 

.118166 

26.38 

15 

.014882 

29.17 

.019400 

29.47 

15 

.114065 

26.00 

.119749 

26.35 

16 

.016632 

29.10 

.021168 

29.42 

16 

.115625 

25.95 

.121330 

26.30 

17 

.018378 

29.05 

.(£2933 

29.35 

17 

.117182 

25.92 

.122908 

26.25 

18 

.020121 

29.00 

.024694 

29.30 

18 

.118737 

25.87 

.124483 

26.22 

19 

.021861 

28.93 

.026452 

29.23 

19 

.120289 

25.82 

.  12GO:)G 

26.17 

20 

.023597 

28.87 

.028206 

29.18 

20 

.121838 

25.77 

.127626 

26.12 

21 

8.025329 

28.82 

8.029957 

29.13 

21 

8.123384 

25.72 

8.129193 

26.07 

22 

.027058 

28.75 

.031705 

29.07 

22 

.124927 

25.68 

.130757 

26.02 

23 

.028783 

28.70 

.033449 

29.00 

23 

.126468 

25.65 

.132318 

25.98 

24 

.030505 

28.65 

.035189 

28.97 

24 

.128006 

25.58 

.133877 

25.93 

25 

.032224 

28.58 

.036927 

28.90 

25 

.129541 

25.55 

.135433 

25.90 

26 

.033939 

28.53 

.038661 

28.83 

26 

.131074 

25.50 

.136987 

25.85 

27 

.a35651 

28.47 

.040391 

28.78 

27 

.132604 

25.45 

.138538 

25.80 

28 

.037:359 

28.42 

.042118 

28.73 

28 

.134131 

25.40 

.140086 

25.75 

29 

.039064 

28.37 

.043842 

28.68 

29 

.135655 

25.37 

.141631 

25.72 

30 

.040766 

28.30 

.045563 

28.62 

30 

.137177 

25.32 

.143174 

25.67 

31 

8.042464 

28.25 

8.047280 

28.57 

31 

8.138696 

25.27 

8.144714 

25.63 

32 

.044159 

28.20 

.048994 

28.50 

32 

.140212 

25.23 

.146252 

25.58 

33 

.045851 

28.13 

.050704 

28.47 

33 

.141726 

25.18 

.147787 

25.53 

34 

.047539 

28.08 

.052412 

28.40 

34 

.143237 

25.13 

.149319 

25.50 

35 

.049224 

28.  as 

.054116 

28.35. 

35 

.144745 

25.10 

.150849 

25.45 

36 

.050906 

27.98 

.055817 

28.28 

•36 

.146251 

25.05 

.152376 

25.40 

37 

.052585 

27.92 

.057514 

28.25 

37 

.147754 

25.02 

.153900 

25.37 

38 

.054260 

27.87 

.059209 

28.18 

38 

.149255 

24.95 

.155422 

25.33 

39 

.055932 

27.82 

.060900 

28.13 

39 

.150752 

24.93 

.156942 

25.27 

40 

.057601 

27.75 

.062588 

28.08 

40 

.152248 

24.88 

.158458 

25.25 

41 

8.059266 

27.72 

8.064273 

28.03 

41 

8.153741 

24.83 

8.159973 

25.18 

42 

.0(50929 

27.65 

.065955 

27.97 

42 

.155231 

24.78 

.161484 

25.17 

43 

.062588 

27.60 

.0676*3 

27.93 

43 

.156718 

24.75 

.162994 

25.10 

44 

.064244 

27.55 

.069309 

27.87 

44 

.158203 

2-1.72 

.164500 

25.07 

45 

.065897 

27.48 

.070981 

27.82 

45 

.159686 

24.67 

.166004 

25.03 

46 

.067546 

27.45 

.072G50 

27.77 

46 

.161166 

24.62 

.167506 

24.98 

47 

.069193 

27.38 

.074316 

27.72 

47 

.162643 

24.58 

.169005 

24.95 

48 

.070836 

27.33 

.075979 

27.67 

48 

.164118 

24.53 

.170502 

24.90 

49 

.072476 

27.30 

.077639 

27.60 

49 

.165590 

24.M 

.171996 

24.87 

50 

.074114 

27.23 

.079295 

27.57 

50 

.167060 

24.45 

.173488 

24.82 

51 

8.07:.748 

27.18 

8.080949 

27.52 

51 

8.168527 

24.42 

8.174977 

24.78 

52 

.077379 

27.13 

.082600 

27.45 

52 

.169992 

24.37 

.176464 

24.73 

53 

.079007 

27.07 

.084247 

27.42 

53 

.171454 

24.33 

.177948 

24.70 

54 

.080631 

27.03 

.085892 

27.37 

54 

.172914 

24.30 

.179430 

24.65 

55 

.082253 

26.98 

.087534 

27.30 

55 

.174372 

24.25 

.180909 

24.  G2 

56 

.083872 

26.93 

.089172 

27.27 

56 

.175827 

24.20 

.182386 

24.58 

57 

.085488 

26.87 

.090808 

27.20 

57 

.177279 

24.17 

.183861 

24.53 

58 

.087100 

26.83 

.092440 

27.17 

58 

.178729 

24.13 

.185333 

24.50 

59 

.088710 

26.78 

.094070 

27.12 

59 

.180177 

24.08 

.186803 

24.47 

GO 

8.090317 

26.72 

8.095697 

27.05  1 

60 

8.181622 

24.05 

8.188271 

24.42 

408 


AND  EXTERNAL  SECANTS. 


L. 

10°                      11° 

Yers. 

D.  1  . 

Ex.  sec. 

D.  I'. 

> 

Vers. 

D.  1". 

Ex.  sec. 

D.  1". 

0 

8.181022 

24.05 

8.188271 

24.42    0  8.264176 

21.85  8.272229  22.27 

i 

.183065 

24.00 

.189736 

24.37 

1 

.265487 

21.82 

.273565 

22.22 

2 

.184505 

23.97 

.191198 

24.35 

2 

.266796 

21.78 

.274898 

22.20 

3 

.185943 

23.93 

.192659 

24.30    3 

.268103 

21.75 

.276230 

22.17 

4 

.187379 

23.88 

.194117 

24.25    4 

.269408 

21.72 

.277560 

22.13 

5 

.188812 

23.85 

.195572 

24.22    5 

.270711 

21.68 

.278888 

22.08 

6 

.190243 

23.80 

.  197025 

24.18  ; 

6 

.272012 

21.65 

.280213 

22.07 

7 

.191671 

23.77 

.198476 

24.15  1 

7 

.273311 

21.62 

.281537 

22.03 

8 

.193097 

23.73 

.199925 

24.10  ! 

8 

.274608 

21.58 

.282859 

22.00 

9 

.194521 

23.68 

.201371 

24.07 

9 

.275903 

21.57 

.284179 

21.98 

10 

.195942 

23.65 

.202815 

24.03 

10 

.277197 

21.52 

.285498 

21.93 

11 

8.197361 

23.62 

8.204257 

23.98 

11 

8.278488 

21.48 

8.286814 

21.90 

12 

.198778 

23.57 

.205696 

23.95 

12 

.279777 

21.47 

.288128 

21.88 

13 

.200192 

23.53 

.207133 

23.92 

13 

.281065 

21.42 

.289441 

21.83 

14 

.201604 

23.50 

.208568 

23.88 

14 

.282350 

21.40 

.290751 

21.82 

13 

.203014 

23.45 

.210001 

23.83 

15 

.283634 

21.37 

.292060 

21.78 

18 

.204421 

23.42 

.211431 

23.80 

16 

.284916 

21.33  1  .293367 

21.75 

17 

.205826 

23.38 

.212859 

23.77 

17 

.286196 

21.28  .294672 

21.72 

18 

.207229 

23.35 

.2142S5 

23.72 

18 

.287473 

21.27   .29597'5 

21.68 

19 

.208630  !  23.30 

.215708 

23.70 

19 

.288749 

21.25   .297276 

21.67 

20 

.2100;>8 

23.27 

.217130 

23.65 

20 

.290024 

21.20 

.298576 

21.62 

21 

8.211424 

23.23 

8.218549 

23.62 

21 

8.291296 

21.17 

8.299873 

21.60 

22 

.212318 

23.18 

.219966 

23.57 

22 

.292566 

21.15 

.301169 

21.57 

23 

.214209 

23.17 

.221380 

23.55 

23 

.293335 

21.10 

.302463 

21.53 

24 

.215599 

23.12 

.222793 

23.50 

24 

.295101 

21.08 

.303755 

21.50 

25 

.216986 

23.08 

.224203 

23.47 

A*O 

.296366 

21.05 

.305045 

21.48 

26 

.218371 

23.03 

.225611 

23.43 

26 

.297629 

21.02 

.306334 

21.43 

27 

.219753 

23.00 

.227017 

23.40 

27 

.298890 

20.98 

.307620 

21.42 

28 

.221133 

22.98 

.223421 

23.35 

28 

.300149 

20.95 

.308905 

21.38 

29 

.222512 

22.93 

.229822 

23.32 

29 

.301406 

20.93 

.310188 

21  .35 

30 

.223888 

22.88 

.231221 

23.30 

30 

.302662 

20.90 

.311469 

21.33 

31 

8.225261 

22.87 

8.232619 

23.25 

31 

,8.303916 

20.85 

8.312749 

21.28 

32 

.226633 

22.82 

.234014 

23.22 

32 

.305167 

20.85 

.314026 

21.27 

33 

.228002 

22  78 

.235407 

23.17 

33 

.306418 

20.80 

.315302 

21.23 

34 

.229369 

23!  77 

.236797 

23.15 

34 

.307666 

20.77 

.316576 

21.22 

35 

.230735 

22.70 

.238186 

23.10 

35 

.308912 

20.75 

.317849 

21  17 

36 

.232097 

22.68 

.239572 

23.08 

36 

.310157 

20.73 

.319119 

21.15 

37 

.2-33458 

22.65 

.240957 

23.03 

37 

.311400 

20.68 

.320388 

21.12 

38 

.234817 

22.60 

.242339 

23.00 

38 

.312641 

20.65 

.321655 

21.08 

39 

.236173 

22.57 

.243719 

22.97 

39 

.313880 

20.62 

.322920 

21.05 

40 

.237'527 

22.55 

.245097 

22.93 

40 

.315117 

20.60 

.324183 

21.03 

41 

8.238880 

22.50 

8.246473 

22.90 

41 

8.316353 

20.57 

8.325445 

21.00 

42 

.2J0230 

22.47 

.247847 

22.87 

42 

.317587 

20.53 

.326705 

20.98 

43 

.241578 

22.43 

.249219 

22.83 

43 

.318819 

20.50 

.327964 

20.93 

44 

.242924 

22.38 

.250589 

22.80 

44 

.320049 

20.48 

.329220 

20.92 

45 

.244267 

22.37 

.251957 

22.75 

45 

.321278 

20.45* 

.330475 

20.88 

46 

.245609 

22.32 

.253322 

22.73 

46 

.322505 

20.42 

.331728 

20.87 

47 

.246948 

22.30 

.254686 

22.68 

47 

.323730 

20.38 

.a32980 

20.82 

48 

.248286 

22.25 

.256047 

22.67 

48 

.324953 

20.37 

.334229 

20.80 

49 

.240321 

22.23 

.257407 

22.62 

49 

.326175 

20.33 

.335477 

20.78 

50 

.250955 

22.18 

.258764  22.60 

50 

.327395 

20.30 

.336724 

20.73 

5i 

8.252286 

22.15 

8.260120 

22.55 

51 

8.  3280  13 

20.27 

3.337968 

20.72 

52 

.253615 

22.12 

.261473 

22.53 

52 

.329829 

20.25 

.339211 

20.70 

53 

.254942 

22.10 

.262825 

22.48 

53 

.331044 

20.22 

.340453 

20.65 

54 

,256268 

22.05 

.264174 

22.47 

54 

.332257 

20.18 

.341692 

20.63 

55 

.257591 

22  02 

.265522 

22.42 

55 

.3a3468 

20.17 

.342930 

20.60 

56 

.258912 

21.98 

.206867 

22.40 

56 

.334678 

20.13 

.344166 

20.58 

57 

.260231 

21.95 

-208211 

22.35 

57 

.335886 

20.10 

.345401 

20.55 

58 

.261548 

21.92 

.269552 

22.33 

58 

.a37092 

20.07 

.346634 

20.52 

59 

.262863 

21.88 

.270892 

22  28 

59 

.3:38296 

20.05 

.,347865 

20.50 

60 

8.264176 

21.85 

8.272229 

22.27  , 

60 

8  339499 

20.02 

8.349095 

20.47 

409 


TABLE  XXVI.— LOGARITHMIC  VERSED   SINES 


12° 

! 

13° 

/ 

Vers. 

D.I'. 

Ex.  sec. 

D.  r. 

/ 

Vers. 

D.  r. 

Ex.  sec. 

D.  r. 

0 

8.339499 

20.02 

8.349095 

20.47 

0 

8.408748 

18.47 

8.420024 

18.95 

1 

.340700 

20.00 

.350323 

20.43 

1 

.409856 

18.43 

.421101 

18.93 

2 

.341900 

19.95 

.351549 

20.42 

2 

.410962 

18.42 

.422297 

18.90 

3 

.343097 

19.95 

.352774 

20.  as 

3 

.412067 

18.40 

.423431 

18.88 

4 

.344294 

19.90 

.353997 

20.35 

4 

.413171 

18.38 

.424564 

18.87 

5 

.345488 

19.88 

.355218 

20.33 

5 

.414274 

18.35 

.425096 

18.83 

G 

.346681 

19.85 

.356438 

20.30 

6 

.415375 

18.32 

.426826 

18.82 

7 

.347872 

19.82 

.357656 

20.28 

7 

.416474 

18.30 

.427955 

18.80 

8 

.349061 

19.80 

.358873 

20.25 

8 

.417572 

18.28 

.429083 

18.77 

9 

.350249 

19.77 

.360088 

20.22 

9 

.418669 

18.25 

.430209 

18.75 

10 

.351435 

19.75 

.361301 

20.20 

10 

.419764 

18.23 

.431334 

18.73 

11 

8.352G20 

19.72 

8.362513 

20.18 

11 

8.420858 

18.22 

8.432458 

18.70 

12 

.353803 

19.68 

.363724 

20.13 

12 

.421951 

18.18 

.433580 

18.67 

13 

.354984 

19.67 

.864932 

20.12 

13 

.423042 

18.17 

.434700 

18.67 

14 

.356164 

19.63 

.366139 

20.10 

14 

.424132 

18.13 

.435820 

18.63 

15 

.357342 

19.60 

.367345 

20.07 

15 

.425220 

18.12 

.436938 

18.62 

16 

.358518 

19.58 

.368549 

20.03 

16 

.426307 

18.10 

.438(tt5 

18.58 

17 

.359693 

19.55 

.369751 

20.02 

17 

.427393 

18.07 

.439170 

18.57 

18 

.360366 

19.53 

.370952 

19.98 

1  18 

.428477 

18.05 

.440284 

18.55 

19 

.362038 

19.50 

.372151 

19.95 

19 

.429560 

18.02 

.44131)7 

18.53 

20 

.363208 

19.48 

.373348 

19.95 

20 

.430641 

18.02 

.442509 

18.50 

21 

8.364377 

19.43 

8.374545 

19.90 

21 

8.431722 

17.97 

8.443619 

18.47 

22 

.335.543 

19.43 

.375739 

19.88 

22 

.432800 

17.97 

.444727 

18.47 

23 

.306709 

19.38 

.376932 

19.85 

j  23 

.433878 

17.93 

.445835 

18.43 

24 

.3G787'2 

19.37 

.378123 

19.83 

24 

.434954 

17.92 

.446941 

18.42 

25 

.369034 

19.35 

.379313 

19.82 

25 

.436029 

17.88 

.448046 

18.38 

26 

.370195 

19.32 

.380502 

19.78 

|  28 

.437102 

17.87 

.449149 

18.38 

27 

.371354 

19.28 

.381689 

19.75 

27 

.438174 

17.85 

.450252 

18.35 

28. 

.37'2511 

19.27 

.382874 

1S.73 

28 

.439245 

17.82 

.451353 

18.32 

29 

.373667 

19.25 

.384058 

19.70 

29 

.440314 

17.80 

.452452 

18.32 

30 

.374822 

19.20 

.385240 

19.68 

30 

.441382 

17.78 

.453551 

18.28 

31 

8.375974 

19.18 

8.386421 

19.65 

31 

8.442449 

17.75 

8.454648 

18.25 

32 

.37/125 

19.17 

.387600 

19.63 

32 

.443514 

17.73 

.455743 

18.25 

33 

.378275 

19.13 

.388778 

19.60 

33 

.444578 

17.72 

.4568:38 

18.22 

34 

.379423 

19.12 

.389954 

19.58 

i  34 

.445641 

17.68 

.457931 

18.20 

35 

.380570 

19.08 

.391129 

19.55 

85 

.446702 

17.68 

.459023 

18.18 

3G 

.381715 

19.05 

.392302 

19.53 

36 

.447763 

17.63 

.460114 

18.15 

37 

.382858 

19.03 

.393474 

19.50 

37 

.448821 

17.63 

.461203 

18.13 

38 

.381000 

19.02 

.394644 

19.48 

38 

.449879 

17.62 

.462291 

18.12 

3;) 

.385141 

18.98 

.395813 

19.45 

39 

.450935 

17.58 

.463:378 

18.10 

40 

.386280 

18.95 

.396980 

19.43 

40 

.451990 

17.55 

.464464 

18.07 

41 

8.387417 

18.93 

8.398146 

19.42 

41 

8.453043 

17.55 

8.465548 

18.05 

42 

.388553 

18.92 

.399311 

19.38 

42 

.454096 

17.52 

.466631 

18.03 

43 

.389688 

18.88 

.400474 

19.35 

43 

.455147 

17.48 

.467713 

18.00 

44 

.390821 

18.85 

.401635 

19.33 

44 

.456196 

17.48 

.4(58793 

18.00 

45 

.391952 

18.83 

.402795 

19.32 

45 

.45/245 

17.45 

.469873 

r.97 

46 

.393082 

18.82 

.403954 

19.28 

46 

.458292 

17.43 

.470951 

r.95 

47 

.394211 

18.78 

.405111 

19.27 

i  47 

.459338 

17.40 

.472028 

r.92 

48 

.395338 

18.75 

.406267 

19.23 

48 

.460382 

17.40 

.473103 

r.9o 

49 

.396463 

18.73 

.407421 

19.22 

49 

.461426 

17.37 

.474177 

r.9o 

50 

.397587 

18.72 

.408574 

19.18 

50 

.462468 

17.35 

.475251 

r.ss 

51 

8.398710 

18.  G8 

8.409725 

19.17 

51 

8.46:5509 

17.32 

8.476322 

17.85 

5.2 

.399831 

18.67 

.410875 

19.13 

62 

.464548 

17.30 

.477393 

r.83 

53 

.400951 

18.63 

.412023 

19.13 

53 

.465586 

17.28 

.478463 

r.so 

54 

.402069 

18.62 

.413171 

19.08 

54 

.466623 

17.27 

.479531 

17.78 

55 

.403186 

18.58 

.414316 

19.08 

i  55 

.467659 

17.23 

.480598 

r.77 

56 

.404301 

18.57 

.415461 

19.03 

i  56 

.468693 

17.23 

.481664 

r.73 

5? 

.405415 

18.53 

.416603 

19.03 

i  57 

.469727 

17.20 

.482728 

1-.73 

58 

.406527 

18.53 

.417745 

19.00 

58 

.470759 

17.17 

.483792 

r.7o 

59 

.407638 

18.50 

.418885 

18.98 

59 

.471789 

17.17 

.484854 

r.68 

60 

8.408748 

18.47 

8.420024 

18.  05 

60 

8.472819 

17.13 

8.485915 

r.67 

410 


AND  EXTERNAL  SECANTS. 


14° 

15° 

' 

Vers. 

D.I". 

Ex.  sec. 

D.I". 

> 

Vers. 

D.I". 

Ex.  sec. 

D.  r. 

0 

8.472819 

17.13 

8.485915 

7.67  : 

0 

8.532425 

15.98 

8.547482 

16.53 

1 

.473847 

17.12 

.486975 

7.63  i 

.533384 

15.97 

.548474 

16.58 

2 

.474874 

17.10 

.4880133 

7.63 

2 

.534342 

15.95 

.549466 

16.52 

8 

.475900 

17.08 

.489091 

7.60  i 

3 

.535299 

15.93 

.550457 

16.50 

4 

.476925 

17.05 

.490147 

7.58  j 

4 

.530255 

15.92 

.551447 

16.48 

5 

.477948 

17.03 

.491202 

7.57  ! 

5 

.537210 

15.88 

.552436 

16.47 

G 

.478970 

17.02 

.492256 

7.53  ! 

6 

.538163 

15.88 

.553424 

10.43 

7 

.479991 

17.00 

.493308 

7.53  i 

7 

.539116 

15.87 

.554410 

10.43 

8 

.481011 

16.97 

.494360 

7.50  : 

8 

.540068 

15.83 

.555396 

16.42 

9 

.482029 

16.95 

.495410 

7  48 

9 

.541018 

15.83 

.556381 

16.38 

10 

.483040 

16.93 

.496459 

7.47  ! 

10 

.541968 

15.80 

.557364 

16.88 

11 

8.484062 

1C.  92 

8.497507 

7.45 

it 

8.542916 

15.78 

8.558347 

16.37 

12 

.485077 

16.90 

.498554 

7.43 

12 

.543863 

15.78 

.559329 

16.33 

13 

.486091 

16.87 

.499GOO 

7.40  ; 

13 

.544810 

15.75 

.5GOE09 

16.33 

14 

.487103 

16.87 

.500644 

7.38  | 

14 

.545755 

15.73 

.561289 

10.30 

15 

.488115 

16.83 

.501087 

7.38 

15 

.546699 

15.72 

.502267 

10.30 

16 

.489125 

16.82 

.502730 

7.35  i 

16 

.547042 

15.70 

.563245 

16.28 

17 

.490134 

16.78 

.503771 

7.32 

17 

.548584 

15.68 

.564222 

16.25 

IB 

.491141 

16.78 

.504810 

7.32 

18 

.549525 

15.67 

.565197 

16.25 

19 

.492148 

16.75 

.505849 

7.30 

19 

.550465 

15.65 

.566172 

16.22 

20 

.493153 

16.73 

.506887 

7.27 

20 

.551404 

15.63 

.567145 

16.22 

21 

8.494157 

16.72 

8.507923 

7.25 

21 

8.552342 

15.62 

8.5G8118 

16.20 

22 

.495160 

16  70 

.508958 

7.25  i 

22 

.553279 

15.00 

.509090 

16.17 

23 

.496162 

16.67 

.509993 

7  22  ! 

23 

.554215 

15.58 

.570060 

16.17 

24 

.497162 

16.67 

.511026 

?!l8  i 

24 

.555150 

15.57 

.571030 

16.15 

25 

.498162 

16.63 

.512057 

7.18  I 

25 

.556084 

15.55 

.571999 

16.12 

26 

.499160 

16.62 

.513088. 

7.17 

26 

.557017 

15.53 

.57'29G6 

16.12 

27 

.500157 

16.60 

.514118 

7.13 

27 

.557949 

15.50 

.573933 

16.10 

28 

.501153 

16.53 

.515146 

7.13 

28 

.558879 

15.50 

.574899 

16.08 

29 

.50:3148 

16.57 

.516174 

7.10 

29 

.559809 

15.48 

.575864 

16.05 

30 

.503142 

16.53 

.517200 

7.08 

30 

.560738 

15.47 

.576827 

16.05 

31 

8.504134 

1G.52 

8.518225 

7.07 

31 

8.561666 

15.43 

8.577790 

10.03 

32 

.505125 

16.52 

.519249 

7.05 

32 

.562592 

15.43 

.578752 

16  C2 

33 

.506116 

16.48 

.520272 

7.03  ! 

33 

.563518 

15  42 

.579713 

16.00 

31 

.507105 

1G.47 

.521294 

7.02  1 

34 

.564443 

15.40 

.580673 

15.  9H 

35 

.508092 

16.47 

.522315 

16.98  i 

35 

.565367 

15.37 

.581632 

15.97 

30 

.509079 

16.43 

.523334 

16.98  ! 

36 

.566289 

15  37 

.582590 

15.95 

37 

.510065 

16.40 

.524353 

16.95  i 

37 

.567211 

15.  a5 

.583547 

15.93 

38 

.511049 

16.40 

.525370 

16.95 

38 

.568132 

15.33 

.584503 

15.92 

39 

.512033 

16.37 

.526387 

16.92 

39 

.569052 

15.30 

.585458 

15.90 

40 

.513015 

16.35 

.527402 

16.90  ! 

40 

.569970 

15.30 

.586412 

15.88 

41 

8.513996 

16.33 

8.528416 

16.88 

41 

8.570888 

15.28 

8.587365 

15.88 

42 

.514976 

10.32 

.529429 

16.87 

42 

.571805 

15.27 

.588318 

15.85 

43 

.515955 

16.28 

.530441 

16.85 

43 

.572721 

15.25 

.589269 

15.83 

44 

.516932 

16.28 

.531452 

16.83 

44 

.573636 

15.22 

.590219 

15.83 

45 

.517909 

16.25 

.532462 

16.82 

45 

.574549 

15.22 

.591169 

15.80 

4(5 

.518884 

16.25 

.533471 

16.78  i 

46 

.575462 

15.20 

.592117 

15.80 

47 

.519859 

16.22 

.534478 

16.78  ! 

47 

.576374 

15.18 

.592065 

15.78 

43 

.520832 

16.20 

.5&>485 

16.75 

48 

.577285 

15.17 

.594012 

15.75 

49 

.521804 

16.18 

.536490 

16.75 

49 

.578195 

15.15 

.594957 

15.75 

50 

.522775 

16.17 

.537495 

16.73 

50 

.579104 

15.13 

.595902 

15.73 

51 

8.523745 

16.15 

8.538498 

16.72 

51 

8.580012 

15.12 

8.596846 

15.72 

52 

.524714 

16.13 

.539501 

16.68 

52 

.580919 

15.10 

.597789 

15.70 

53 

.525682 

16.10 

.540502 

16.67 

53 

.581825 

15.08 

.598731 

15.68 

54 

.526648 

16.10 

.541502 

16.65 

54 

.582730 

15.07 

.599C7'2 

15.67 

55 

,527'614 

16.07 

.542501 

16.63 

55 

.583634 

15.05 

.000612 

15.05 

5G 

.528578 

16.07 

.543499 

16.63 

56 

.584537 

15.05 

.601551 

15.65 

57 

.529542 

16.03 

.544497 

16.60  i 

57 

.585440 

15.02 

.  602490 

15.62 

58 

.5305'  14 

16.02 

.545493 

16.58  i 

58 

.586341 

15.00 

.603427 

15.00 

59 

.531465 

16.00 

.546488 

16.57  ! 

59 

.587241 

15.00 

.604363 

15.60 

60 

8.532425 

15.98 

8.547483 

16.53 

60 

8.588141 

14.97 

8.605299 

15.58 

411 


TABLE  XXVI.-LOGARITHMIC  VERSED  SINES 


16° 

17° 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  r. 

' 

Vers. 

D.  1". 

Ex.  sec. 

D.  r. 

0 

8.58S141 

14.97 

8.605299 

15.58 

0 

8.640434 

14.08  8.659838 

14.72 

i 

.589339 

14.95 

.606234 

15.55 

1 

.641279 

14.07 

.660721 

14.72 

2 

.589936 

14.95 

.607167 

15.5.3 

2 

.642123 

14.05 

.661604 

14.70 

3 

.591)333 

14.93 

.603100 

15.53 

!  3   .642966 

14.05 

.662486 

14.68 

4 

.591729 

14.93 

.603032 

15.52 

4   .643809 

14.02 

.663367 

14.68 

5 

.592523 

14.90 

.609933 

15.50 

5  1  .644650 

14.02 

.664248 

14.65 

6 

.593517 

14.88 

.610333 

15.50 

6 

.645491   14.00 

.665127 

14.65 

7 

.594410 

14.87 

.611823 

15.47 

7 

.646331 

13.98 

.665006 

14.63 

8 

.595302 

14.83 

.612751 

15.45 

8 

.647170 

13.97 

.666884 

14.62 

9 

.595192 

14.83 

.613378 

15.45 

9 

.648008 

13.95 

.667761 

14.60 

10 

.597082 

14.82 

.614605 

15.43 

10 

.648845 

13.95 

.6G8G37 

14.60 

11 

8.597971 

14.82 

8.615331 

15.42 

11 

8.649632 

13.93 

8.669513 

14.58 

12 

.593350 

14.78 

.616456 

15.38 

12 

.6.30518 

13.92 

.670388 

14.57 

13 

.599747 

14.77 

.617379 

15.38 

13 

.651353 

13.90 

.671262 

14.55 

14 

.603333 

14.75 

.618302 

15.33 

14 

.652187 

13.88 

.672135 

14.55 

15 

.601518 

14.75 

.619225 

15.35 

15 

.653020 

13.87 

.673003 

14.52 

16 

.6024)3 

14.72 

.620146 

15.33 

16 

.653852 

13.87 

.673379 

14.52 

17 

.633233 

14.72 

.621053 

15.33 

17 

.654684 

13.85 

.674750 

14.50 

18 

.604163 

14.70 

.621933 

15.30 

18 

.  653515 

13.83 

.675620 

14.50 

19 

.633331 

14.67 

.622334 

15.30 

19 

.653345 

13.82 

.676490 

14.47 

2J 

.635931 

14.67 

.62J322 

15.38 

20 

.657174 

13.84 

.677358 

14.47 

•  21 

8.633311 

14.65 

8.621739 

15.27 

21 

8.658003 

13.78 

8.678226 

14.45 

22 

.637633 

14.63 

.623335 

15.25 

22 

.658330 

13.78 

.679093 

14.45 

2} 

.633333 

14.62 

.623370 

15.23 

23 

.659557 

13.77 

.679960 

14.42 

24 

_633445 

14.60 

.627434 

15.23 

24 

.680483 

13.75 

.680825 

14.42 

23 

.610321 

14.60 

.628393 

15.20 

25 

.661338 

13.73 

.681690 

14.40 

23 

.611197 

14.57 

.629310 

15.20 

23 

.662132 

13.73 

.682354 

14.38 

27 

.612371 

14.57 

.630222 

15.18 

27 

.662356 

13.72 

.683417 

14.38 

23 

.612315 

14.53 

.63113} 

15.17 

28 

.663779 

13.70 

.684280 

14.35 

23 

.613317 

14.53 

.632313 

15.15 

29 

.664601 

13.63 

.685141 

14.35 

33 

.6141533 

14.52 

.632332 

15.13 

30 

.665422 

13.67 

.686002 

14.35 

31 

8.615563 

14.50 

8.633350 

15.13 

31 

8.666242 

13.67 

8.686S63 

14.32 

33 

.616433 

14.43 

.634763 

15.10 

32 

.667032 

13.65   .687722 

14.32 

33 

.617293 

14.47 

.635574 

15.10 

33 

.667881 

13.63 

.688581 

14.30 

31 

.61816? 

14.45 

.633330 

15.08 

34 

.663699 

13.62 

.689439 

14.28 

33 

.619034 

14.45 

.637485 

15.07 

35 

.639516 

13.60 

.690296 

14.28 

33 

.619331 

14.42 

.6'33»3 

15.05 

36 

.670332 

13.60 

.691153 

14.25 

37 

.623766 

14.42 

.633232 

15.05 

37 

.671148 

13.58 

.692008 

14.25 

33 

.621631 

14.40 

.640195 

15.02 

33 

.671963 

13.57 

.692863 

14.25 

33 

.622495 

14.33 

.641035 

15.02 

39 

.672777 

13.55 

.693718 

14.22 

40 

.623353 

14.37 

.641937 

15.00 

40 

.673590 

13.55 

.694571 

14.22 

41 

8.624220 

14.35 

8.642397 

14.98 

41 

8.674403 

13.53 

8.695424 

14.20 

'42 

.625031 

14.33 

.643793 

14.97 

42   .675215 

13.52 

.696276 

14.18 

43 

.625941 

14.33 

.644694 

14.95 

43   .676026 

13.50  !  .697127 

14.18 

41 

.6-26801 

14.30 

.645591 

14.95 

44   .676836 

13.48  j  .697978 

14.17 

45 

.627659 

14.30 

.64648S 

14.93 

45 

.677(545 

13.48   .698828 

14.15 

46 

.6-28517 

14.23 

.647334 

14.92 

43 

.678154 

13.47 

.699677  !  14.13 

47 

.629374 

14.27 

.648279 

14.90 

47 

.679262 

13.45   .700525  14.13 

48 

.630233 

14.25 

.649173   14.83 

48 

.680069 

13.43 

.701373 

14.12 

49 

.631035 

14.23 

.650333  1  14.87 

49 

.680875 

13.43 

.702220 

14.10 

50 

.631939 

14.22 

.650958 

14.87 

50 

.681681 

13.42 

.703066 

14.10 

51 

8.632792 

14.22 

8.651850 

14.85 

51 

8.6S2486 

13.40 

8.703912 

14.07 

52 

.633645 

14.18 

.65-2741 

14.83 

52 

.683290 

13.38 

.704756 

14.07 

53 

.634495 

14.18 

.653631 

14.82 

53  '  .684093 

13.38 

.705600 

14.07 

54 

.635347 

14.17 

.654520 

14.80 

54   .684896 

13.35 

.706444 

14.03 

55 

.636197 

14.15 

.655408 

14.80 

55   .685697 

13.35 

.707286 

14.03 

56 

.637046 

14.13 

.656296 

14.77 

56 

.686498 

13.35 

.708128  14.02 

57 

.637894 

14.13 

.657182 

14.77 

57   .6S7'299 

13.32 

.708969  14.02 

38 

.638742 

14.10 

.658068 

14.77 

58  !  .6881)98 

13.32 

.709810  14.00 

59 

.639588 

14.10 

.6.38954 

14.73   59   .688897 

13.30 

.710650  13.98 

60 

8.640434 

14.08 

8.659838 

14.72  II  60  8.68)695 

13.23  i8.  71  1489  13.97 

412 


AND  EXTERNAL  SECANTS. 


18° 

19° 

/ 

Vers. 

D.  1'. 

Ex.  sec.  D,  1". 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  1'. 

0  8.089695  !  13.28 

8.711489 

13.97 

0 

8.730248 

12.58 

8.760578 

13.30 

1 

690492  13.28 

.712327 

13.95 

1 

.737003 

12.57 

.761376 

13.30 

2 

.691289 

13.25 

.713104 

13.95 

2 

.737757 

12.55 

.762174 

13.28 

3 

.692084 

13.25 

.714001 

13.95 

3 

.738510 

12.55 

.762971 

13.27 

4 

.692879 

13.25 

.714838 

13.92 

4 

.739263 

12.53 

.763767 

13.27 

5 

.693674 

13.22 

.715673 

13.92 

5 

.740015 

12.52 

.764563 

13  25 

6 

.694467 

13.22 

.716508 

13.90 

6 

.740766 

12  ..50 

.765358 

13.23 

7 

.695260 

13.20 

.717342 

13.88 

7 

.741516 

12.50 

.766152 

13.23 

8 

.696052 

13.18 

.718175 

13.88 

8 

.742266 

12.50 

.766946 

13.22 

9 

.696843 

13.18 

.719008 

13.87 

9 

.743016 

12.47 

.767739 

13.20 

10 

.697634 

13.17 

.719840 

13.85 

10 

.743764 

12.47 

.768531 

13.20 

11 

8.698424 

13.15 

8.720671 

13.85 

11 

8.744512 

12.45 

8.769323 

13.18 

12 

.699213 

13.13 

.721502 

13.83 

12 

.745259 

12.43 

.770114 

13.18 

13 

.700001 

13.13 

.722332 

13.82 

13 

.746006 

12.45 

.770905 

13.17 

14 

.700789 

13.12 

.723161 

13.80 

14 

.746752 

12.42 

.771695 

13.15 

15 

.701576 

13.10 

.723989 

13.80 

15 

.747497 

12.42 

.772484 

13.15 

16 

.702362 

13.08 

.724817 

13.78 

16 

.748242 

12.40 

.773273 

13.13 

17 

.703147 

13.08 

.725644 

13.78 

17 

.748986 

12.38 

.774061 

13.13 

18 

.703932 

13.07 

.726471 

13.77 

18 

.749729 

12.38 

.774849 

13.12 

19 

.704716 

13.05 

.727297 

13.75 

19 

.750472 

12.37 

.775636 

13.10 

20 

.705499 

13.05 

.728122 

13.73 

30 

.751214 

12.35 

.776422 

13.08 

21 

8.706282 

13.02 

8.728946 

13.73 

21 

8.751955 

12.35 

8.777207 

13.10 

22 

.707063 

13.  (2 

.729770 

13.72 

22 

.752696 

12.33 

.777993 

13.07 

23 

.707844 

13.02 

.730593 

13.70 

23 

.753436 

12.32 

.778777 

13.07 

24 

.708625 

12.98 

.731415 

13.70 

24 

.754175 

12.32 

.779561 

13.05 

25 

.709404 

12.98 

.732237 

13.68 

25 

.754914 

12.30 

.780344 

13.05 

26 

.710183 

12.98 

.7,33058 

13.67 

26 

.755652 

12.28 

.781127 

13.03 

27 

.710961 

12.95 

.7,33878 

13.67 

27 

.756389 

12.28 

.781909 

13.02 

28 

.711739 

12.95 

.734698 

13.65 

28 

.757126 

12.27 

.782690 

13.03 

29 

.712516 

12.93 

.735517 

13.63 

29 

.757862 

12.27 

.7&3471 

13.00 

30 

.713292 

12.92 

.736886 

13.63 

30 

.758598 

12.25 

.784251 

13.00 

31 

8.714067 

12.92 

8.737153 

13.62 

31 

8.759333 

12.23 

8.785031 

12.98 

32 

.714842 

12.90 

.737970 

13.60 

32 

.760067 

12.23 

.785810 

12.97 

33 

.715016 

12.88 

.738786 

13.00 

33 

.760801 

12.22 

.786588 

12.97 

34 

.716889 

12.87 

.739602 

13.58 

34 

.761534 

12.20 

.787366 

12.97 

35 

.717161 

12.87 

.740417 

13.57 

35 

.762266 

12.20 

.788144 

12.93 

36 

.717988 

12.85 

.741231 

13.57 

36 

.762998 

12.18 

.788920 

12.93 

37 

.718704 

12.85 

.742045 

13.55 

!  37 

.763729- 

12.17 

.789696 

12  93 

88 

.719475 

12.82 

.742858 

13.53 

38 

.764459 

12.17 

.790472 

12.92 

39 

.720214 

12.82 

.743070 

13.53  i  39 

.765189 

12.15 

.791247 

12.90 

40 

.721013 

12.82 

.744482 

13.52  |  40 

.765918 

12.15 

.792021 

12.90 

41 

8.721782 

12.78 

8.745293 

13.50 

41 

8.766647 

12.12 

8.792795 

12.88 

42 

.722549 

12.78 

.746103 

13.50 

42 

.767374 

12.13 

.793568 

12.87 

43 

.  728316 

12.78 

.746913 

13.48 

43 

.768102 

12.10 

.794340 

12.87 

44 

.724083 

12.75 

.747722 

13.47 

44 

.768828 

12.10. 

.795112 

12.87 

45 

.724848 

12.75 

.748530 

13.47 

45 

.769554 

12.10 

.795884 

12.83 

46 

.725013 

12.73 

.749338 

13.45 

46 

.770280 

12.08 

.796654 

12.85 

47 

.726377 

12.72 

.750145 

13.43 

47 

.771005 

12.07 

.797425 

12.82 

48 

.727140 

12.72 

.750951 

13.43 

48 

.771729 

12.05 

.798194 

12.82 

49 

.727903 

12.70 

.751757 

13.42 

49 

.772452 

12.05 

.798963 

12.82 

50 

.728665 

12.70 

.752562 

13.42 

50 

.773175 

12.05 

.799732 

12.80 

51 

8.729427 

12.67 

8.75:3367 

13.40 

51 

8.773898 

12.02 

8.800500 

12.78 

52 

.730187 

12.67 

.754171 

13.38 

52 

.774619 

12.02 

.801267 

12.78 

53 

.730947 

12.67 

.7541)74 

13.37 

53 

.775340 

12.02 

.802034 

12.77 

54 

.731707 

12.63 

.755776 

13.37 

51 

.770061 

12.00 

.802800 

12.75 

55 

.732405 

12.63 

.756578 

13.37 

55 

.776781 

11.98 

.803565 

12.75 

56 

.733223 

12.  63 

.757380 

13.33 

50 

.777500 

11.97 

.804330 

12.75 

57 

.7*3981 

12.  00 

.758180 

13.33  |  57 

.778218 

11.97 

.805095 

12.73 

58 

.784787 

12.00 

.758980 

13.33  I  58 

.778936 

11.97 

.805859 

12.72 

59 

.735493 

12.58 

.759780 

13.30   59 

.779654 

11.93 

.806622 

12.72 

60 

8.736248 

12.58 

8.760578 

13.30  l|  60 

8.780370 

11.95 

8.807385 

12.70 

413 


TABLE  XXVI.-LOGARITHMIC  VERSED    SINES 


20° 

21° 

/ 

Vers. 

D.  r. 

Ex.  sec. 

D.  1'. 

' 

Vers. 

D.  r. 

Ex.  sec.  D.  1". 

~0~ 

8.780370 

11.95  8.807385   12.70 

0 

8.822296 

11.35 

8.852144  i  12.17 

1 

.781087 

11.92 

.808147  12.68 

1 

.822977 

11.35 

.852874 

12.17 

2 

.781802 

11.92 

.808908 

12.68 

2 

.823658 

11.33 

.853604 

12.13 

3 

.782517 

11.90 

.809669 

12.68 

3 

.824338 

11.33 

.854*32  12.15 

4 

.783231 

11.90 

.810430 

12.67 

4 

.825018 

11.32   .855061  I  12.13 

5 

.783945 

11.88 

.811190 

12.65 

5 

.825697 

11.32   .855789  12.12 

G 

.784658 

11.88 

.811949 

12.65 

6 

.826375 

11.30   .856516 

12.12 

7 

.785371 

11.87 

.812708 

12.63 

7 

.827051 

11.28 

.857243  !  12.10 

8 

.786083 

11.85 

.813466 

12.63 

8 

.82',  731 

11.28 

.857969   12.10 

9 

.786794 

11.85 

.814224 

12.62 

9 

.828408 

11.28 

.858695   12  08 

10 

.787505 

11.83 

.814981 

12.60 

10 

.829085 

11.27 

.859430  12.08 

11 

8.788215 

11.82 

8.815737 

12.60 

11 

8.829761 

11.25 

8.860145  12.  0V 

12 

.788924 

11.82 

.816493 

12.60 

12 

.830436 

11.25 

.860869  12.07 

13 

.7896-33 

11.82 

.817249 

12.58 

13 

.831111 

11.23 

.861593  12.05 

14 

.790342 

11.78 

.818004 

12.57 

14 

.831785 

11.23 

.862316  1  12.05 

15 

.791049 

11.78 

.818758 

12.57 

!  15 

.832459 

11.22 

.863039   12.03 

16 

.791756 

11.78 

.819512 

12.55 

16 

.833132 

11.20 

.863761  !  12.03 

17 

.792463 

11.77 

.820265 

12.55 

17 

.833804 

11.20 

.864483  j  12.02 

18 

.793169 

11.75 

.821018 

12.53 

18 

.834476 

11.20 

.865204   12.02 

19 

.793874 

11.75 

.821770 

12.52 

19 

.8.35148 

11.18 

.865925   12.02 

20 

.794579 

11.73 

.822521 

12.52 

20 

.835819 

11.17 

.866646  |  11.98 

21 

8.795283 

11.73 

8.823272 

12.52 

21 

8.836489 

11.17 

8.867365 

12.00 

22 

.795987 

11.72 

.824023 

12.50 

i  22 

.837159 

11.17 

.868085 

11.98 

23 

.796690 

11.70 

.824773 

12.48 

i  23 

.837829 

11.15 

.868804  11.97 

24 

.797392 

11.70 

.825522 

12.48 

24 

.838498 

11.13 

.869522   11.97 

25 

.798094 

11.68 

.826271   12.47 

25 

.839166 

11.13 

.870240  11.95 

26 

.798795 

11.68 

.827019  12.47 

26 

.839834 

11.12 

.870957 

11.  £5 

27 

.799496 

11.67 

.827767 

12.45  j  27 

.840501 

11.12 

.871674   11.93 

28 

.800196 

11.67 

.828514 

12.45  i  28 

.841168  j  11.10 

.872390  |  11.93 

29 

.800896 

11.63 

.82-261   12.43  ||  29 

.841834 

11.10 

.873106  i  11.  S3 

30 

.801594 

11.65 

.830007  12.42 

30 

.842500 

11.08 

.873822 

11.62 

31 

8.802293 

11.63 

8.830753 

12.42  i  31 

8.843165 

11.07 

8.874537 

11.90 

32 

.802991 

11.62 

.831497  12.42  1  32 

.843829 

11.07 

.875251   11.90 

33 

.803688 

11.60 

.832242  j  12.40  li  33 

.844493 

11.07 

.875965  11.88 

34 

.804384 

11.60 

.832986  !  12.38  ||  34 

.845157 

11.05 

.876678  i  11.88 

35 

.805080 

11.60 

.833729 

12.38  ii  35 

.845820  11.05 

.877391  ;  11.88 

36 

.805776 

11.58 

.834472 

12.38  ||  36 

.846483  11.03 

.878104  11.87 

37 

.806471 

11.57 

.835815 

12.37  !!  37 

.847145 

11.02 

.878816 

11.87 

38 

.807165 

11.57 

.835957 

12.35  i  38 

.847806 

11.02 

.879528 

11.85 

39 

.807859 

11.55 

.836698 

12.35  ji  39 

.848467 

11.00 

.880239 

11.83 

40 

.808552 

11.53 

.837439 

12.33   40 

.S49127 

11.00 

.880949 

11.83 

41 

8.809244 

11.53 

8.838179 

12.33 

41 

8.849787 

11.00 

8.881659 

11.83 

42 

.809936 

11.53 

.838919 

12.32 

42 

.850447 

10.98 

.882369 

11.82 

43 

.810628 

11.52 

.639058 

12.30 

43 

.851106 

10.97 

.883078 

11.82 

44 

.811319 

11.50   .840396 

12.32 

44 

.851764 

10.97 

.883787 

11.  SO 

45 

.812009 

11.50   .841135 

12.28   45 

.852422 

10.95 

.884495 

11.80 

46 

.812699 

11.48   .841872  12.28  !  46 

.853079 

10.95 

.885203 

11.78 

47 

.813388 

11.48   .842609   12.28  j  47 

.853736 

10.93 

.885910 

11.78 

48 

.814077 

11.47   .843346  12.27   48 

.854392 

10.93   .886617  i  11.77 

49 

.814765 

11.45   .844082 

12.25   49 

.855048 

10.92  '•  .887323   11.77 

5'J 

.815452 

11.45   .844817  12.25  ||  50 

.855703 

10.92   .888029 

11.75 

51 

8.816139 

11.43 

8.845552  12.25   51 

8.856358 

10.90  !8.  888734 

11.75 

52 

.816825 

11.43 

.846287 

12.23 

52 

.a57012 

10.90 

.889439 

11.75 

53 

.817511 

11.42 

.847021 

12.22   53 

.857666 

10.88 

.890144 

11.73 

54 

.818196 

11.42 

.847754 

12.22   54 

.858319 

10.88 

.890848 

11.72 

55 

.818881 

11.40 

.848487 

12.22   55 

.858972 

10.87 

.891551 

11.72 

56 

.819565 

11.40 

.849220 

12.20  !  56   .859634 

10.87 

.892254 

11.72 

57 

.820249 

11.38 

.849952 

12.18   57   .860276 

10.85 

.892957 

11.70 

58 

.820932 

11.37 

.850683 

12.18  |j  58 

.860937 

10.  &5 

.893659 

11.70 

59 

.821614 

11.37 

.851414 

12.17   59   .861578 

10.83 

.894361 

11.68 

60  8.822296  11.35  8.852144  12.171  6018.862228 

10.82  8.895062  11.68 

414 


ANL>   EXTERNAL  SECANTS. 


22°                      23° 

f 

Vers. 

D.  r. 

Ex.  sec. 

D.  1". 

\  > 

Vers. 

D.  r. 

Ex.  sec.;D.  1". 

0  8.862228  10.82 

8.895062  11.68 

o 

8.900341 

10.33  8.936315   11.23 

1 

.862877   10.83 

.895763  11.67 

1 

.900961 

10.  35 

.936989  11.23 

2 

.863527  10.80 

.896463  11.67 

1  2 

.901582 

10.32 

.937663  I  11.22 

3 

.864175  10.80 

.897163 

11.65 

3   .902201 

10.  33 

.938336  ;  11.22 

4 

.864823 

10.80 

.897862 

11.65 

4 

.902821 

10.32 

.939009  11.22 

5 

.865471 

10.78 

.898561   11.63 

5 

.903440   10.30 

.939682  11.20 

6 

.866118 

10.  rs 

.899259  11.63 

6 

.904058 

10.30 

.940354   11.20 

7 

.86676') 

10.77 

.899957   11.63  ||  7 

.904676 

10.28 

.941026   11.20 

8 

.867411 

10.77 

.900655   11.62 

8 

.905293 

10.28 

.941698   11.18 

9 

.868057 

10.75 

.901352   11.60 

9 

.905910 

10.28 

.942369  i  11.17 

10 

.868702 

10.73 

.902048 

11.62 

10 

.906527 

10.27 

.943039  (  11.  Id 

11 

8.869346 

10.75 

8.902745 

11.58 

11 

8.907143 

10.27 

8.943710  11.15 

12 

.869991 

10.72 

.903440 

11.60 

12 

.907759 

10.25   .944379 

11.17 

13 

.870634 

10.72 

.904136 

11.57 

13 

.908:374 

10.25   .945049 

11.15 

14 

.871277 

10.72 

.904830  11.58 

14 

.908989 

10.23  i  .945718   11.18 

15 

.871920 

10.70 

.905525   11.57 

15 

.909603 

10.23   .946386 

11.13 

16 

.872.562 

10.70 

.906219 

11.55 

16 

.910217 

10.22   .947054 

11.13 

17 

.873204 

10.68 

.906912 

11.55 

17 

.910830 

10.22  i  .947722 

11.12 

18 

.873845 

10.68 

.907605 

11.53 

18 

.911443 

10.22  !  .948389 

11.12 

19 

.874486 

10.67 

.908298 

11.53 

19 

.912056 

10.20   .949056  11.12 

20 

.875123 

10.67 

.908990 

11.52 

20 

.912668 

10.18   .949723  11.10 

21 

8.875766 

10.65 

8.909681 

11.52 

21 

8.913279 

10.20  8.950389 

11.10 

22 

.876405 

10.65 

.910372 

11.52 

22 

.913891 

10.17 

.951055 

11.08 

23 

.877044 

10.63 

.911063 

11.52 

1  23 

.914501 

10.17 

.951720 

11.08 

24 

.877682 

10.63 

.911754 

11.48 

24 

.915111 

10.17  |  .952385 

11.07 

25 

.878320 

10.62 

.912443 

11.50 

25 

.915721 

10.17  j  .953049 

11.07 

23 

.878957 

10.62 

.913133 

11.48 

!  26 

.916331 

10.15 

.953713 

11.07 

27 

.879594 

10.60 

.913822 

11.47 

|  27 

.916940 

10.13 

.954377 

11.05 

23 

.880230 

10.60   .914510 

11.47 

28 

.917548 

10.13   .955040 

11.05 

29 

.8808® 

10.58   .915198 

11.47 

29 

.918156 

10.13   .955703 

11.05 

30 

.881501 

10.58   .915886   11.45 

30 

.918764 

10.12   .956366 

11.03 

31 

8.882136 

10.58  !  8.916573 

11.45 

31 

8.919371 

10.10  ^8.  957028 

11.03 

32 

.882771 

10.57   .917260 

11.43 

32 

.919977   10.12 

.957690  ;  11.02 

33 

.883405 

10.55 

.917946 

11.43 

88 

.920584 

10.10 

.958351 

11.02 

34 

.8840:38 

10.  55 

.918632 

11.43 

i  34 

.921190 

10.  (?8  i  .959012 

11.00 

35 

.884671 

10.53 

.919318 

11.42 

35 

.921795 

10.08   .959672 

11.00 

36 

.885:303 

10.53 

.920003 

11.40 

36 

.922400 

10.07   .960332 

11.00 

37 

.885935 

10.53 

.920687 

11.42 

i  37   .923004 

10.07 

.960992   10.98 

38 

.886567 

10.52 

.921372 

11.38 

1  33   .923608 

10.07 

.9616.-)! 

10.98 

39 

.887198 

10.52 

.922055  11.40 

39 

.924212 

10.05   .962310 

10.98 

40 

.887829 

10.50   .922739 

11.37 

40 

.9.24815 

10.05  i  .962969  10.97 

41 

8.888459 

10.48  8.923421 

11.38 

41 

8.925418 

10.03  ^  8.  963627 

10.97 

42 

.889088 

10.48   .924104 

11.37 

42 

.926020 

10.03 

.964285 

10.95 

43  l  .889717 

10.48   .924786 

11.  a5 

43 

.926622 

10.03 

.964942 

10.95 

44  i  .890346 

10.47   .925467 

11.37 

44 

.927224 

10.02 

.965599   10.95 

45 

.890974 

10.47   .926149 

11.33 

45 

.927825 

10.00 

.966256 

10.93 

46 

.891602 

10.45   .926829 

11.35 

46 

.928425 

10.00 

.966912 

10.93 

47   .8922.29 

10.45   .927510 

11.32 

47 

.929025 

10.00 

.967568 

10.92 

48 

.892856 

10.43 

.928189 

11.33 

48 

.939625 

9.98 

.968223 

10.92 

49 

.893482 

10.48 

.928869 

11.32 

49 

.930224 

9.98 

.968878  i  10.92 

50 

.894108 

10.42 

.929548 

11.30 

50 

.930823 

9.9? 

.969533  j  10.90 

51 

8.894733 

10.42 

.8.930226 

11.32 

51 

8.931421 

9.97 

8.970187  10.90 

52  |  .895358  10.42 

.930905 

11.28 

52 

.932019 

9.97 

.970841  i  10.88 

53 

.895983   10.40   .931582 

11.30 

53 

.932617 

9.95 

.971494 

10.88 

54 

.896607  10.38 

.932260 

11.27 

!  54 

.933214 

9.95 

.972147 

10.88 

55 

.897230  ;  10.38 

.932936 

11.28 

55 

.933811 

9.93 

.972800 

10.87 

56 

.897853  '  10.38 

.933613 

11.27 

56 

.934407 

9.93 

.973452 

10.87 

57  i  .898476   10.37 

.934289 

11.27   57 

.935003 

9.92 

.974104 

10.87 

58  j  .899098 

10.35 

.934965 

11.25   58 

.9a5598  1  9.92 

.974756  j  10.85 

59  ,  .899719 

10.37 

.935640 

11.25  1  59 

.936193   9.92 

.975407  10.85 

60  ;  8.900341  i  10.  :i3  !  8.936315 

11.23  i!  60  8.936788  !  9.90 

8.976058  I  10.83 

415 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


24° 

25° 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  r. 

/ 

Vers. 

D.I". 

Ex.  sec. 

i 

D.  r. 

0  \  8.936788 

9.90 

8.976058  10.83 

0  8.971703 

9.50 

9.014428 

i  10.47 

1 

.937382 

9.90 

.976708 

10.83 

1 

.972273 

9.48 

.015056 

10.48 

2 

.937976 

9.88 

.977358  I  10.83 

2 

.972842 

9.48 

.015685 

!  10.45 

3 

.938569 

9.88 

.978008  10.82 

3 

.973411 

9.48 

.016312 

<  10.47 

4 

.939162   9.87 

.978657  10.82 

4 

.973980  9.47 

.016940 

1  10.45 

5 

.939754   9.87 

.979306  10.80 

5 

.974548 

9.47 

.017567 

10.45 

6 

.940346   9.87 

.979954  10.80 

6 

.975116 

9.45 

.018194 

10.45 

7 

.940938 

9.85 

.980602  10.80 

7 

.975683 

9.45 

.018821 

10.43 

8 

.941529 

9.85 

.981250 

10.80 

8 

.976250 

9.43 

.019447 

i  10.43 

9 

.942120 

9.83 

.981898 

10.78 

9 

.976816 

9.43 

.020073 

i  10.42 

10 

.942710 

9.83 

.982545 

10.77 

10 

.977382 

9.43 

.020698 

10.42 

11 

8.943300 

9.82 

8.9&3191 

10.77 

11 

8.977948 

9.43 

9.021323 

10.42 

12 

.943889 

9.83 

.983837 

10.77 

12 

.978514 

9.42 

.021948 

10.40 

13 

.944479 

9.80 

.984483 

10.77 

13 

.979079 

9.40 

.022573 

10.42 

14 

.945067 

9.80 

.985129 

10.75 

14 

.979643 

9.40 

.023197 

10.38 

15 

.945655 

9.80 

.985774 

10.75 

15 

.980207 

9.40 

.023820 

10.40 

16 

.946243 

9.80 

.986419 

10.73 

16 

.980771 

9^40 

.024444 

10.38 

17 

.946831 

9.78 

.987063 

10.73 

17 

.981386 

9.38 

.025067 

10.38 

18 

.947418 

9.77 

.987707 

10.73 

18 

.981898 

9.37 

.025690 

10.37 

19 

.948004 

9.77 

.988351 

10.72 

19 

.982460 

9.38 

.026313 

10.37 

20 

.948590 

9.77 

.988994 

10.72 

20 

.983023 

9.37 

.026934 

10.37 

21 

8.949176* 

9.75 

8.989637 

10.70 

21 

8.983585 

9.35 

9.027556 

10.  a5 

22 

.949761 

9.75 

.990279 

10.72 

22 

.984146 

9.35 

.028177 

10.35 

23 

.950346 

9.75 

.990922 

10.68 

23 

.984707 

9.35 

.028798 

10.35 

24 

.950931 

9.73 

.991563 

10.70 

24 

.985268 

9.38 

.029419 

10.83 

25 

.951515 

9.73 

.992205 

10.68 

25 

.985838  9.  -33 

.030039 

10.33 

26 

.952099 

9.72 

.992846 

10.68 

26 

.986388  9.33 

.031)659 

10.33 

27 

.952682 

9.72 

.993487 

10.67 

27 

.986948  9.32 

.031279 

10.33 

28 

.953265 

9.70 

.994127 

10.67 

28 

.987507  9.32 

.031899 

10  32 

29 

.953847 

9.70 

.994767 

10.65 

29 

.988066  9.32 

.032518 

10.30 

30 

.954429 

9.70 

.995406 

10.67 

30 

.988625  !  9.30 

.033136 

10.28 

31 

8.955011 

9.68 

8.996046 

10.65 

31 

8.989183  9.28 

9.033755 

10.30 

32 

.955592 

9.68 

.996685 

10.63   32 

.989740 

9.30 

.034373 

10.30 

33 

.956173 

9.67 

.997323 

10.63 

33 

.990298 

9.28 

.034991 

10.28 

34 

.956753 

9.68 

.997961 

10.  63  ! 

34 

.990855 

9.27 

.Oa5608 

10.28 

35 

.957334 

9.65 

.998599 

10.62  j 

35 

.991411 

9.28 

.036225 

10.28 

36 

.957913 

9.65 

.999236 

10.62 

36 

.991968 

9.25 

.036842 

10.27 

37 

.958492 

9.65 

8.999873 

10.62 

37 

.992523 

9.27 

.037458 

10.27 

38 

.959071 

9.65 

9.000510 

10.60 

38 

.993079 

9.25 

.038074 

10.27 

39 

.959650 

9.63 

.001146 

10.62 

39 

.993634 

9.25 

.038(590 

10.  as 

40 

.960228 

9.62 

.001783 

10.58 

40 

.994189 

9.23 

.039305 

10.25 

41 

8.960805 

9.62 

9.002418 

10.58 

41 

8.994743 

9.23 

9.039920 

10.25 

42 

.961382 

9.62 

.003053 

10.58 

42 

.995297 

9.23 

.0405:35 

10.25 

43 

.961959 

9.60 

.003688 

10.58 

43 

.995851 

9.22 

.041150 

10.23 

44 

.962535 

9.60 

.004323 

10.57 

44 

.996404 

9.22 

.011764 

10.23 

45 

.963111 

9.60 

.004957 

10.57 

.45 

.996957 

9.20 

.042378 

10.22 

46 

.963687 

9.58 

.005591 

10.55 

46 

.997509 

9.22 

.042991 

10.22 

47 

.964262 

9.58 

.006224 

10.57 

47 

.998062 

9.18 

.043604 

10.22 

48 

.964837 

9.57 

.006858 

10.53 

48 

.998613 

9.20 

.044217 

10.22 

49 

.965411 

9.57 

.007490 

10.55 

49 

.999165 

9.18 

.044830 

10.20 

50   .965985 

9.57 

.008123 

10.53 

50 

8.999716 

9.17 

.045442 

10.20 

51  8.966559 

9.55 

9.008755 

10.53 

51 

9.000266 

9.18 

9.046054 

10.18 

52 

.967132 

9.55 

•009387 

10.52 

52 

.000817 

9.17 

.046665 

10.18 

53 

.967705 

9.53 

.010018 

10.52 

53 

.001367 

9  15 

.047276 

10.18 

54 

.968277 

9.53 

.010649  1  10.52 

54 

.001916 

9.17 

.047887 

10.18 

55 

.968849 

9.53 

.011280 

10.50 

55 

.002466 

9.13 

.048498 

10.17 

56 

.969421 

9.52 

.011910 

10.50 

56 

.003014 

9.15 

.049108 

10.17 

57 

.969992 

9.52 

.012540 

10.50 

57 

.003563 

9.13 

.049718 

10.17 

58 

.970563 

9.50 

.013170 

10.48  ! 

58 

.004111 

9.13 

.050328 

10.15 

59 

.971ia3 

9.50 

.013799 

10.48 

59 

.004659  9.12 

.030937 

10.15 

60 

8.971703 

9.50 

9.014428 

10.47 

60 

9.005206 

9.12 

9.051546 

10.15 

416 


AND  EXTERNAL  SECANTS. 


26° 

27° 

' 

Vers. 

D.  1". 

Ex.  sec. 

D.  1". 

' 

Vers. 

D.I". 

Ex.  sec.  D.  1". 

o 

9.005206 

9.12 

9.051546 

10.15 

0 

9.037401 

8.77 

9.087520   9.83 

1 

.005753 

9.12 

.052155 

10.13 

1 

.037927 

8.75 

.088110   9.83 

2 

.006300 

9.10 

.05276:3 

10.13 

2 

.038452 

8.77 

.088700   9.83 

3   .006846 

9.10 

.053371 

10.13 

3 

.038978 

8.75 

.089290   9.83 

4 

.007392 

9.10 

.053979 

10.12 

4 

.039503 

8.73 

.089880  i  9.82 

5 

.007938 

9.08 

.054586 

10.12 

5 

.040027 

8.75 

.090469   9.82 

6 

.0084&3 

9.08 

.055193 

10.12 

6 

.040552 

8.73 

.091058 

9.82 

7   .009028 

9.07 

.055800 

10.10 

7 

.041076 

8.72 

.091647 

9.80 

8   .009573 

9.07 

.056406 

10.10 

8 

.041599 

8.73 

.092235 

9.80 

9 

.010116 

9.07 

.057012 

10.10 

9 

.042123 

8.72 

.092823 

9.80 

10 

.010600 

9.05 

.057618 

10.10 

10 

..042646 

8.70 

.093411 

9.78 

11 

9.011203 

9.05 

9.058224 

10.08 

11 

9.043168 

8.72 

9.093998 

9.80 

18 

.011746 

9.05 

.058829 

10.08 

12 

.043691 

8.70 

.094586 

9.78 

13 

.012289 

9.03 

.059434 

10.07 

13 

.044213 

8.70 

.095173 

9.77 

14 

.012831 

9.03 

.060638 

10.08 

14 

.044735 

8.68 

.095759 

9.78 

15 

.013373 

9.03 

.060643 

10.07 

15 

.045256 

8.68 

.096346 

9.77 

16 

.013915 

9.02 

.061247 

10.05 

16 

.045777 

8.68 

.096^32 

9.77 

17 

.014456 

9.02 

.061850 

10.07 

17 

.046298 

8.67 

.097518 

9.75 

18 

.014997 

9.02 

.062454 

10.05 

18 

.046818 

8.67 

.098103 

9.77 

19 

.015538 

9.00 

.063057 

10.03 

19 

.047*38 

8.67 

.098689 

9.75 

20 

.016078 

9.00 

.063659 

10.05 

20 

.047858 

8.65 

.099274 

9.73 

21 

9.016618 

8.98 

9.064262 

10.03 

21 

9.048377 

8.65 

9.099a58 

9.75 

22 

.017157 

9.00 

.064864 

10.03 

22 

.048896 

8.65 

.100443 

9.73 

23 

.917697 

8.97 

.065466 

10.02 

23 

.049415 

8.63 

.101027 

9.73 

21 

.018235 

8.98 

.066067 

10.02 

24 

.049933 

8.63 

.101611 

9.72 

86 

.018774 

8.97 

.066668 

10.02 

25 

.050451 

8.63 

.102194 

9.73 

26 

.019312 

8.97 

.067269 

10.02 

26 

.050969 

8.63 

.102778 

9.72 

.019850 

8.95 

.067870 

10.00 

i  27 

.051487 

8.62 

.103361 

9.70 

28 

.020387 

8.95 

.068470 

10.00 

28 

.052004 

8^60 

.103943 

9.72 

29 

.020924 

8.95 

.069070 

10.00 

29 

.052520 

8.62 

.104526 

9.70 

30 

.021461 

8.93 

.069670 

9.98 

30 

.053037 

8.60 

.105108 

9.70 

31 

9.021997 

8.93 

9.070269 

9.98 

31 

0.053553 

8  60 

9.105690 

9.68 

88 

.022533 

8.93 

.070868 

9.98 

32 

.054069 

8  58 

.106271 

9.70 

33 

.023069 

8.92 

.071467 

9.97 

33 

.054584 

8.58 

.106853 

9.68 

84 

.023604 

8.92 

.072065 

9.97 

34 

.055099 

8.58 

.107434  |  9.68 

85 

.024139 

8.90 

.072663 

9.97 

35 

.055614 

8.58 

.108015   9.67 

85 

.024673 

8.92 

.073261 

9.97 

36 

.056129 

8.57 

.108695 

9.67 

37 

.025208 

8.90 

.073859 

9.95 

37 

.056643 

8.57 

.109175 

9.67 

38 

.025742 

8.88 

.074456 

9  95 

j  38 

.057157 

8.55 

.109755 

9.67 

39 

.026275 

8.88 

.075053 

9.93 

39 

.05767'0 

8.55 

.110335 

9.65 

40 

.026808 

8.88 

.075649 

9.95 

40 

.058183 

8.55 

.110914 

9.67 

41 

9.027341 

8.88 

9.076246 

9.93 

41 

9.058696 

8.55 

9.111494 

9.63 

42 

.027874 

8.87 

.076842 

9.92 

42 

.059209 

8.53 

.112072 

9.65 

43 

.028406 

8.87 

.077437 

9.93 

43 

.059721 

8.53 

.112651 

9.63 

41 

-.028938 

8.85 

.078033 

9.92 

44 

.0602:33 

8.53 

.113229 

9.63 

45 

.029469 

8.85 

.078628 

9.92 

45 

.060745 

8.52 

.113807 

9.63 

46 

.030000 

8.85 

.07'9223 

9.90 

46 

.061256 

8.52 

.114385 

9.63 

47 

.0:50531 

8.85 

.079817 

9.92  1  47 

.0617'67 

8.50 

.114963 

9.62 

43 

.031062 

8.83 

.080412 

9.90  1  48 

.062277 

8.52 

.115540 

9.62 

49 

.031592 

8.83 

.081006 

9.88  |  49 

.062788 

8.50 

.116117 

9.60 

50 

.032122 

8.82 

.081599 

9.90   50 

.0(53298 

8.48 

.116693 

9.62 

51 

9.032651 

8.82 

9.0S2193 

9.88   51 

9.063807 

8.50 

9.117270 

9.60 

52 

.033180 

8.82 

.082786 

9.87   52 

.064317 

8.48 

.117846 

9.60 

53 

.033709 

8.80 

.083378 

9.88  1  53 

.064826 

8.48 

.118422 

9.58 

54 

.034237 

8.80 

.083971 

9.87 

54 

.065335 

8.47 

.118997 

9.60 

55 

.084765 

8.80 

.084563 

9.87 

55 

.065843 

8.47 

.119573 

9.58 

56 

.035293 

8.78 

.085155 

9.87 

56 

.066351 

8.47 

.120148 

9.58 

57 

.Oa5820 

8.78 

.085747 

9.85 

57 

.066859 

8.45 

.120723 

9.57 

58 

.036347 

8.78 

.086338 

9.85   58 

.067366 

8.47 

.121297 

9.57 

59 

.036874 

8.78 

.086929 

9.85   59 

.067874 

8.43 

.121871 

9.57 

60  9.037401 

8.77 

9.087520 

9.83  II  60 

9.068380 

8.45 

9.122445 

9.57 

417 


TABLE  XXVI.— LOGARITHMIC  VERSEb  SINES 


28C 

29° 

' 

Vers. 

D.  r. 

Ex.  sec. 

D.  1'. 

' 

Vers. 

D.  r. 

Ex.  sec. 

D.  1". 

0 

9.068380 

8.45 

9.122445 

9.57 

1  0 

9.098229 

8.15 

9.156410 

9.30 

1 

.068887 

8.43 

.123019 

9.57 

|  1 

.098718 

8.13 

.156968 

9.32 

2 

.069393 

8.43 

.123593 

9.55 

2 

.099206 

8.12 

.157527 

9.28 

3 

.069899 

8.43 

.124166 

9.55 

3 

.099693 

8.13 

.158084 

9.30 

4 

.070405 

8.42 

.124739 

9.53 

4 

.100181 

8.12 

.158642 

9.30 

5 

.070910 

8.42 

.125311 

9.55 

I  5 

.100668 

8.12 

.159200 

9.28 

6 

.071415 

8.40 

.125884 

9.53 

6 

.101155 

8.12 

.159757 

9.28 

7 

.071919 

8.42 

.126456 

9.53 

7 

.101642 

8.10 

.160314 

9.27 

8 

.072424 

8.40 

.127028 

9.52 

8 

.102128 

8.10 

.160870 

9.28 

9 

.072928 

8.40 

.127599 

9.53 

9 

.102614 

8.10 

.161427 

9.27 

10 

.073432 

8.38 

.128171 

9.52 

10 

.103100 

8.08 

.161983 

9.27 

11 

9.073935 

8.38 

9.128742 

9.52 

11 

9.103585 

8.08 

9.162539 

9.27 

12 

.074438 

8.38 

.129313 

9.50 

12 

.  104070 

8.08 

.163095 

9.25 

13 

.074941 

8.37 

.129883 

9.50 

13 

.104555 

8.08 

.163650 

9.25 

14 

.075443 

8.38 

.  130453 

9.50 

14 

.105040 

8.07 

.164205 

9.25 

15 

.075946 

8.35 

.131023 

9.50 

1  15 

.  103324 

8.07 

.164760 

9.25 

16 

.076447 

8.37 

.131593 

9.50 

16 

.106008 

8?05 

.165315 

9.25 

17 

.076949 

8.35 

.132163 

9.48 

17 

.106491 

8.07 

.165870 

9.23 

18 

.077450 

8.35 

.132732 

9.48 

18 

.106975 

8.05 

.166424 

9.23 

19 

.077951 

8.35 

.133301 

9.48 

19 

.107458 

8.05 

.16697'8 

9.23 

20 

.078452 

8.33 

.133870 

9.47 

20 

.107941 

8.03 

.167532 

9.22 

21 

9.078952 

8.33 

9.134438 

9.47 

21 

9.108423 

8.05 

9.168085 

9.23 

22 

.079452 

8.33 

.135008 

9.47 

22 

.108908 

8.03 

.168639 

9.22 

23 

.079952 

8.32 

.135574 

9.47 

1  23 

.109388 

8.02 

.169192 

9.22 

24 

.030451 

8.32 

.136142 

9.45 

24 

.109869 

8.03 

.169745 

9.20 

25 

.080950 

8.32 

.136709 

9.47 

25 

.110351 

8.02 

.170297 

9.22 

26 

.081449 

8.32 

.137277 

9.45 

26 

.110832 

8.02 

.170850 

9.20 

27 

.081948 

8.30 

.1*7844 

9.43 

27 

.111313 

8.00 

•  .171402 

9.20 

28 

.082446 

8.30 

.138410 

9.45 

28 

.111793 

8.  CO 

.171954 

9.18 

29 

.082944 

8.28 

.138977 

9.43 

29 

.112273 

8.00 

.172505 

9.20 

30 

.083441 

8.30 

.139543 

9.43 

30 

.112753 

8.00 

.173057 

9.18 

31 

9.083939 

8.28 

9.140109 

9.42 

31 

9.1132.33 

8.00 

9.173608 

9.18 

32 

.084436 

8.27 

.140674 

9.43 

32 

.113713 

7.98 

.174159 

9.18 

33 

.084932 

8.28 

.141240 

9.42 

33 

.114192 

7.98 

.  74710 

9.17 

34 

'  .085429 

8.27 

.141805 

9.42 

34 

.114671 

.97 

.  75260 

9.17 

35 

.085925 

8.25 

.142370 

9.40 

35 

.115149 

.97 

.175810 

9.17 

36 

.086420 

8.27 

.142934 

9.42 

36 

.115627 

.97 

.176360 

9.17 

37 

.086916 

8.25 

.143199 

9.40 

37 

.116105 

.97 

.176910 

9.17 

38 

.087411 

8.25 

.144063 

9.40 

38 

.116583 

.97 

.  77460 

9.15 

39 

.087906 

8.23 

.144827 

9.38 

39 

.117061 

.95 

.178009 

9.15 

40 

.088400 

8.25 

.  145190 

9.40 

40 

.117538 

.95 

.178558 

9.15 

41 

9.088895 

8.23 

9.145754 

9.38 

I  41 

9.118015 

.93 

9.179107 

9.15 

42 

.089389 

8.22 

.146317 

9.38 

42 

.118491 

.95 

.179656 

9.13 

43 

.089882 

8.23 

.146880 

9.37 

43 

.118968 

.93 

.180204 

9.13 

44 

.090376 

8.22 

.147442 

9.38 

44 

.119444 

.92 

.180752 

9.13 

45 

.090869 

8.22 

.148005 

9.37 

45 

.119919 

.90 

.181300 

9.13 

46 

.091362 

8.20 

.148567 

9.37 

46 

.120395 

.92 

.181848 

9.12 

47 

.091854 

8.20 

.149129 

9.35 

47 

.120870 

.92 

.182395 

9.13 

48 

.092346 

8.20 

.149690 

9.35 

48 

.121345 

.92 

.182943 

9.12 

49 

.092838 

8.20 

.150251 

9.37 

49 

.121820 

.90 

.183490 

9.10 

50 

.093330 

8.18 

.150813 

9.33 

|  50 

.122294 

.90 

.184036 

9.12 

51 

9.093821 

8  18 

9.151373 

9.35 

i  51 

9.122768 

.90 

9.184583 

9.10 

52 

094312 

8.18 

.151934 

9.33  l 

52 

.123242 

.88 

.185129 

9.10 

53 

094803 

8.17 

.  152494 

9.35 

53 

.  1237  15 

.90 

.185675 

9.10 

54 

.095293 

8.17 

.153055 

9.32 

54 

.124189 

.88 

.186221 

9.10 

55 

.095783 

8.17 

.153614 

9.33 

55 

.124662 

.87 

.186767 

9.08 

56 

.096273 

8.17 

.154174 

9.32 

56 

.125134 

.88 

.187312 

9.10 

57 

.096763 

8.15 

.154733 

9.33 

57 

.125607 

.87 

.187'858 

9.08 

58 

.097252 

8.15 

.155293 

9.30 

58 

.126079 

.87 

.188403 

9.07 

59 

097741 

8.13 

.155851 

9.32 

59 

.126551 

.85 

.188947 

9.08 

60 

9.098229 

8.15 

9.156410 

9.30 

60 

9.127022 

.87 

9.189492 

9.07 

418 


AND  EXTERNAL  SECANTS. 


30° 

31° 

/ 

Vers. 

D.  r. 

Ex.  sec. 

D.  1'. 

/ 

Vers. 

D.  r. 

Ex.  sec. 

D.  1'. 

0 

9.127022 

.87 

9.189492 

9.07 

0 

9.154828 

7.58 

9.221762 

8.85 

1 

.127494 

.85 

.190036 

9.07 

1 

.1552813 

7.58 

.222293 

8.87 

2 

.127965 

.85 

.190580 

9.07 

2 

.155738 

7.58 

.222825 

8.83 

3 

.128436 

.83 

.191124 

9.07 

3 

.156193 

7.58 

.223355 

8.85 

4 

.128906 

.83 

.191668 

9.05 

4 

.156648 

7.57 

.223886 

8.R5 

5 

.129376 

.83 

.192211 

9.05 

5 

.157102 

7.57 

.224417 

8.83 

6 

.129846 

.83 

.192754 

9.05 

G 

.157556 

7.57 

.224947 

8.83 

.130316 

.82 

.193297 

9.05 

7 

.158010 

7.57 

.225477 

8.83 

8 

.130785 

.83 

.193840 

9.03 

8 

.158464 

7.55 

.226007 

8.83 

9 

.131255 

.82 

.194382 

9.05 

9 

.158917 

7.55 

.226537 

8.82 

10 

.131724 

7.80 

.194925 

9.03 

10 

.159370 

7.55 

.227066 

8.82 

11 

9.132192 

7.80 

9.195467 

9.03 

11 

9.159823 

7.55 

9.227595 

8.83 

12 

.132660 

7.82 

.196009 

9.02 

12 

.160276 

7.53 

.228125 

8.80 

13 

.133129 

7.78 

.196550 

9.03 

13 

.160728 

7.53 

.228653 

8.82 

14 

.133596 

7.83 

.197092 

9.02 

14 

.161180 

7.53 

.229182 

8.82 

15 

.134064 

7.78 

.197633 

9.02 

15 

.161632 

7.52 

.229711 

8.80 

16 

.13-1531 

7.78 

.198174 

9.02 

16 

.162083 

7.53 

.230239 

8.80 

17 

.134998 

7.78 

.198715 

9.00 

17 

.162535 

7.52 

.230767 

8.80 

18 

.135465 

7.77 

.199255 

9.00 

18 

.162986 

7.52 

.231295 

8.78 

19 

.135931 

7.77 

.199795 

9.00 

19 

.163437 

7.50 

.231822 

8.80 

20 

.136397 

7.77 

.200335 

9.00 

20 

.163887 

7.52 

.232350 

8.78 

21 

9.136863 

7.77 

9.200875 

9.00 

21 

9.164338 

7.50 

9.232877 

8.78 

22 

.137329 

7.75 

.201415 

8.98 

22 

.164788 

7.48 

.233404 

8.78 

23 

.137794 

7.77 

.201954 

9.00 

23 

.165237 

7.50 

.  .233931 

8.78 

24 

.138260 

7.73 

.202494 

8.97 

24 

.165687 

7.48 

.234458 

8.77 

25 

.138724 

7.75 

.203032 

8.98 

25 

.166186 

7.48 

.234984 

8.77 

26 

.139189 

7.73 

.203571 

8.98 

26 

.166585 

7.48 

.235510 

8.77 

27 

.139653 

'7.73 

.204110 

8.97 

27 

.167034 

7.48 

.236036 

8.77 

28 

.140117 

7.73 

.204648 

8.97 

28 

.167483 

7.47 

.236562 

8.77 

29 

.140581 

7.73 

.205186 

8.97 

29 

.167931 

7.47 

.237088 

8.75 

30 

.141045 

7.72 

.205724 

8.97 

30 

.168379 

7.47 

.237613 

8.77 

31 

9.141508 

7.72 

9.206262 

8.95 

31 

9.168827 

7.47 

9.238139 

8.75 

3-2 

.141971 

7.72 

.206799 

8.97 

32 

.169275 

7.45 

.238664 

8.75 

33 

.142434 

7.70 

.207:337 

8.95 

33 

.169722 

7.45 

.239189 

8.73 

3-1 

.142896 

7.70 

.207874 

8.93 

34 

.170169 

7.45 

.239713 

8.75 

35 

.143358 

7.70 

.208410 

8.95 

35 

.170616 

7.43 

.240238 

8.73 

36 

.143820 

7.70 

.208947 

8.93 

36 

.171062 

7.45 

.240762 

8.73 

O^ 

.1442*3 

7.68 

.209483 

8.95 

37 

.171509 

7.43 

.241286 

8.73 

38 

.144743 

7.68 

.210020 

8.93 

38 

.171955 

7.42 

.241810 

8.72 

39 

.145204 

7.68 

.210556 

8.92 

39 

.172400 

7.43 

.242333 

8.73 

40 

.145665 

7.68 

.211091 

8.93 

40 

.172846 

7.42 

.242857 

8.72 

41 

9.146126 

7.67 

9.211627 

8.92 

41 

9.173291 

7.42 

9.243380 

8.72 

42 

.146586 

7.67 

.212162 

8.92 

42 

.173736 

7.42 

.243903 

8.72 

43 

.147046 

7.67 

.212697 

8.92 

43 

.174181 

7.42 

.244426 

8.72 

44 

.147506 

7.67 

.213232 

8.92 

44 

.174626 

7.40 

.244949 

8.70 

45 

.147966 

7.65 

.213767 

8.90 

45 

.175070 

7.40 

.245471 

8.72 

46 

.148425 

.65 

.214301 

8.92 

46 

.175514 

7.40 

.245994 

8.70 

47 

.148884 

.65 

.214836 

8.90 

47 

.175958 

7.40 

.246516 

8.70 

48 

.149:343 

.63 

.215370 

8.90 

48 

.176402 

7.38 

.247'038 

8.68 

49 

.149801 

.63 

.215904 

8.88 

49 

.176845 

7.38 

.247559 

8.70 

50 

.150259 

.63 

.216437 

8.90 

50 

.177288 

7.38 

.248081 

8.68 

51 

9.150717 

.63 

9.216971 

8.88 

51 

9.177731 

7.38 

9.248602 

8.68 

52 

.151175 

.63 

.217504 

8.88 

52 

.178174 

7.37 

.249123 

8.68 

53 

.151633 

M 

.218037 

8.88 

53 

.178616 

7.37 

.249644 

8.68 

54 

.152090 

.62 

.218570 

8.87 

54 

.179058 

7.37 

.250165 

8.68 

55 

.152547 

.60 

.219102 

8.88 

55 

.179500 

7.37 

.250686 

8.67 

56 

.153003 

.62 

.219635 

8.87 

56 

.179942 

7.S5 

.251206 

8.67 

57 

.153460 

.60 

.220167 

8.87 

57 

.180383 

7.37 

.251726 

8.67 

58 

153916 

.60 

.220699 

8.87 

58 

.180825 

7.33 

.252246 

8.67 

59 

.154372 

.60 

.221231 

8.85 

59 

.181265 

7.35 

.252766 

8.67 

60 

9.154828 

7.58 

9.221762 

8.85 

1  60 

9.181706 

7.35 

9.253286 

8.65 

419 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


32° 

33° 

/ 

Vers. 

D.I". 

Ex.  sec. 

D.r. 

/ 

Vers. 

D.r. 

Ex.  sec. 

D.r. 

o 

9.181706 

7.35 

9.25328(5 

8.65 

0 

9.207714 

7.10 

9.284122 

8.48 

1 

.182147 

7.33 

.258805 

8.65 

1 

.208140 

.10 

.284631 

8.47 

2 

.182587 

7.33 

.254324 

8.05 

2 

.208566 

.10 

.285139 

8.47 

3 

.183027 

7.32 

.254843 

8.65 

3 

.208992 

.10 

.285047 

8.47 

4 

.183466 

7.33 

.255302 

8.05 

4 

.209418 

.08 

.280155 

8.47 

5 

.183906 

7.32 

.255881 

8.03 

5 

.209843 

.08 

.280063 

8.45 

6 

.184345 

7.32 

.256399 

8.05 

6 

.210208 

7.08 

.287170 

8.47 

7 

.184784 

7.32 

.256918 

8.03 

7 

.210093 

7.08 

.287078 

8.45  ! 

8 

.185223 

7.32 

.257436 

8.03 

8 

.211118 

7.08 

.288185 

8.45 

9 

.185682 

7.80 

.257954 

8.02 

9 

.211543 

7.07 

.288092 

8.45 

10 

.186100 

7.30 

.258471 

8.03 

!  10 

.211907 

7.07 

.289199 

843 

11 

9.186538 

7.30 

9.258989 

8.62 

11 

9.212391 

7.07 

9.289705 

8.45 

12 

.186976 

7.28 

.259506 

8.62 

12 

.212815 

7.07 

.290212 

8.43 

13 

.187413 

7.30 

.260023 

8.  '62 

13 

.213239 

7.05 

.290718 

8.43 

14 

.187851 

7.28 

.260540 

8.62 

14 

.213062 

7.05 

.291224 

8.43 

15 

.188288 

7.27 

.261057 

8.62 

i  15 

.214085 

7,05 

.291730 

8.43 

10 

.188724 

7.28 

.261574 

8.60 

16 

.214508 

7.05 

.292230 

8.43 

17 

.189161 

7.27 

.262090 

8.00 

17 

.214931 

7.05 

.292742 

8.42 

18 

.189597 

7.28 

.202606 

8.60 

18 

.215354 

7.03 

.293247 

8.43 

19 

.190034 

7.25 

.263122 

8.00 

19 

.215776 

7.03 

.25)3753 

8.42 

20 

.190469 

7.27 

.263638 

8.00 

20 

.216198 

7.03 

.294258 

8.42 

21 

9.190905 

7.27 

9.264154 

8.58 

21 

9.216620 

7.03 

9.294763 

8.42 

22 

.191341 

7.25 

.264669 

8.58 

22 

.217042 

7.02 

.295268 

8.40 

23 

.191776 

7.25 

.265184 

8.00 

23 

.217403 

7.02 

.295772 

8.42 

24 

.192211 

7.23 

.266700 

8.57 

24 

.217884 

7.02 

.290277 

8.40 

25 

.192645 

7.25 

.266214 

8.58 

25 

.218305 

7.02 

.290781 

8.40 

26 

.193080 

7.23 

.266729 

8.58 

26 

.218720 

7.00 

.297285 

8.40 

27 

.  194514 

7.23 

.267244 

8.57 

27 

.219140 

7.02 

•  .297789 

8.40 

28 

.193948 

7.23 

.267758 

8.57 

28 

.219567 

7.00 

.298293 

8.40 

29 

.194382 

7.22 

.268272 

8.57 

29 

.219987 

7.00 

.298797 

8.38 

30 

.194815 

7.23 

.268786 

8.57 

30 

.220407 

0.98 

.299300 

8.38 

31 

9.195249 

7.22 

9.269300 

8.57 

31 

9.220826 

7.00 

9.299803 

8.40 

32 

.195682 

7.22 

.269814 

8.55 

32 

.221246 

0.98 

.300307 

8.37 

33 

.196115 

7.20 

.270327 

8.55 

!  33 

.221005 

0.98 

.£00809 

8.38 

34 

.196547 

7.22 

.270840 

8.57 

!  M 

.222084 

0.98 

.301312 

8.38 

35 

.196980 

7.20 

.271354 

8.53 

35 

.222503 

0.97 

.201815 

8.37 

36 

.197412 

7.20 

.271866 

8.55 

36 

.222921 

0.98 

.302317 

8.38 

37 

.197844 

7.18 

.272379 

8.55 

37 

.223340 

0.97 

.£02820 

8.37 

38 

.198275 

7.20 

.272892 

8.53 

38 

.223758 

0.97 

.303322 

8.37 

39 

.198707 

7.18 

.273404 

8.53 

39 

.224176 

6.95 

.  £03824 

8.35 

40 

.199138 

7.18 

.273916 

8.53 

40 

.224593 

6.97 

.204325 

8.37 

41 

9.199569 

7.18 

9.274428 

8.53 

41 

9.225011 

6.95 

9.204827 

8.35 

42 

.200000 

7.17 

.274940 

8.53 

42 

.225428 

6.95 

.205328 

8.37 

43 

.200430 

7.18 

.275452 

8.52 

43 

.225845 

6.95 

.805880 

8.85 

44 

.200861 

7.17 

.275963 

8.52 

44 

.220202 

0.93 

.200331 

8.35 

45 

.201291 

7.15 

.270474 

8.53 

45 

.220078 

0.95 

.306832 

8.35 

46 

.201720 

7.17 

.270986 

8.50 

46 

.227095 

0.93 

.307333 

8.33 

47 

.202150 

7.15 

.277496 

8.52 

47 

.227511 

0.93 

.307833 

8.35 

48 

.202579 

7.15 

.278007 

8.52 

48 

.227927 

6.92 

.308334 

8.33 

49 

.203008 

7.15 

.278518 

8.50 

49 

.228342 

6.93 

.308834 

8.33 

50 

.203437 

7.15 

.279028 

8.50 

50 

.228758 

6.92 

.309334 

8.33 

51 

9.203866 

7.13 

9.279538 

8.50 

51 

9.229173 

6.92 

9.309834 

8.33 

52 

.204294 

7.15 

.280048 

8.50 

52 

.229588 

0.92 

.310334 

8.33 

53 

.204723 

7.13 

.280558 

8.50 

53 

.230003 

0.92 

.310834 

8.22 

54 

.205151 

7.12 

.281088. 

8.48  ! 

54 

.230418 

0.90 

.311883 

8.32 

55 

.205578 

7.13 

.281577 

8.50  | 

55 

.230832 

6.90 

.311832 

8.32 

56 

.206006 

7.13 

.282087 

8.48 

56 

.281246 

6.90 

.312331 

8.32 

57 

.2064*3 

7.12 

.282590 

8.48  | 

57 

.231000 

6.90 

.312830 

8.32 

58 

.206860 

7.12 

.283105 

8.48 

58 

.232074 

6.88 

.313329 

8.32 

59 

.207287 

7.12 

.283614 

8.47 

59 

.232487 

6.90 

.313828 

8.30 

60 

9.207714 

7.10 

9.284122 

8.48  i 

60 

9.232901 

6.88 

9.314326 

8.32 

420 


AND  EXTERNAL  SECANTS. 


34° 

| 

35° 

1 

Vers. 

D.  1". 

Ex.  sec. 

D.  1".  I 

/ 

Vers 

D  r. 

Ex.  sec. 

D.  1". 

0 

9.23-3301 

6.88 

9.31432(5 

8.32 

0 

9.257314 

6.67 

9.343949 

8.15 

1 

.233314 

6.88 

.314825 

8.30  ! 

1 

.257714 

6.68 

.344438 

8.15 

2 

.233727 

6.87 

.315323 

8.30  i 

2 

.258115 

6.67 

.344927 

8.15 

3 

.234139 

6.88 

.315821 

8.30 

3 

.258515 

6.67 

.345416 

8.13 

4 

.234552 

6.87 

.316319 

8.30 

4 

.258915 

6.65 

.345904 

8.15 

5 

.234934 

6.87 

.316817 

8.28  i 

5 

.259314 

6.67 

.346393 

8.13 

6 

.235376 

6.87 

.317314 

8.28  i 

6 

.259714 

6.65 

.346881 

8.13 

7 

.235783 

6.85 

.317811 

8.30  ! 

r- 

.260113 

6.65 

.347369 

8.13 

8 

.23(5199 

6.87 

.318309 

8.28  i 

8 

.260512 

C.65 

.347857 

8.13 

9 

.233311 

6.85 

.318300 

8.28  l 

9 

.260911 

6.65 

.348345 

8.13 

10 

.237022 

6.85 

.319:303 

8.27  | 

10 

.261310 

6.65 

.348833 

8.13 

11 

9.237433 

6.85 

9.319799 

8.28  ! 

11 

9.261709 

6.63 

9.349321 

8.12 

12 

.237844 

6.83 

.320290 

8.27  ! 

12 

.262107 

6.63 

.349808 

8.12 

13 

.238254 

6.85 

.320792 

8.28  i 

13 

.262505 

6.63 

.350295 

8.12 

14 

.2331505 

6.83 

.321289 

8.27  1 

14 

.262903 

6.63 

.350782 

8.12 

15 

.239375 

6.83 

.321785 

8.27  i 

15 

.263301 

6.62 

.351269 

8.12 

16 

.239485 

6.82 

.322281 

8.25 

16 

.263698 

6.63 

.351756 

8.12 

17 

.239394 

6.83 

.322776 

8.27 

17 

.264096 

6.62 

.352243 

8.12 

18 

.240304 

6.82 

.323272 

8.27 

18 

.264493 

6.62 

.352730 

8.10 

19 

.240713 

6.82 

.323763 

8.25 

19 

.264890 

6  62 

.353216 

8.10 

20 

.211122 

6.82 

.324263 

8.25 

20 

.265287 

6.60 

.353702 

8.10 

21 

9.241531 

6.82 

9.324758 

8.25 

21 

9.265683 

6.62 

9.a54188 

8.10 

22 

.241940 

6.82 

.325253 

8.25 

22 

.266080 

6.60 

.354674 

8.10 

23 

.242348 

6.80 

.325748 

8.25 

23 

.266476 

6.60 

.355160 

8.10 

24 

.242756 

6.80 

.326243 

8.23 

24 

.266872 

6.58 

.355646 

8.08 

23 

.2431(54 

6.80 

.326737 

8.25 

25 

.267267 

6.60 

.356131 

8.10 

28 

.243572 

6.80 

.327232 

8.23 

26 

.267663 

6.58 

.a56617 

8.08 

27 

.243930 

6.78 

.327723 

8.23 

27 

.268058 

6.58 

.357102 

8.08 

23 

.2413S7 

6.78 

.323220 

8.23 

28 

.268453 

6.58 

.357587 

8.08 

29 

.244794 

6.78 

.323714 

8.22 

29 

.268848 

6.58 

.358072 

8.08 

30 

.215.201 

6.78 

.329207 

8.23 

30 

.269243 

6.58 

.358557 

8.08 

31 

9.245(>03 

6.7? 

9.329701 

8.23 

31 

9.269638 

6.57 

9.359042 

8.07 

32 

.246014 

6.78 

.330195 

8.22 

32 

.270032 

6.57 

.359526 

8.08 

33 

.245421 

6.77 

.330383 

8.22 

33 

.270426 

6.57 

.360011 

8.07 

34 

.24(5327 

6.77 

.&31181 

8.22 

34 

.270820 

6.57 

.360495 

8.07 

35 

.247233 

6.77 

.331674 

8.22 

35 

.271214 

6.57 

.360979 

8.07 

30 

.247633 

6.75 

.332167 

8.20 

36 

.271608 

6.55 

.361463 

8.07 

37 

.243944 

6.75 

.332659 

8.22 

37 

.272001 

6.55 

.361947 

8.07 

33 

.243449 

6.75 

.333152 

8.20 

38 

.272394 

6.55 

.362431 

8.05 

39 

.248854 

6.75 

.333644 

8.22 

39 

.272787 

6.55 

.362914 

8.07 

40 

.249259 

6.75 

.334137 

8.20 

40 

.273180 

6.53 

.363398 

8.05 

41 

9.249064 

6.73 

9.334629 

8.20 

41 

9.273572 

6.55 

9.363881 

8.05 

42 

.250003 

6.75 

.335121 

8.18 

42 

.273965 

6.53 

.364364 

8.05 

43 

.250473 

6.73 

.335612 

8.20 

43 

.274357 

6.53 

.364847 

8.05 

44 

.250377 

6.73 

.336104 

8.18 

44 

.274749 

6.£3 

.365330 

8.05 

45 

.251281 

6.72 

.336595 

8.20  j 

45 

.275141 

6.52 

.365813 

8.03 

46 

.251034 

6.73 

;  337087 

8.18 

46 

.275532 

6.53 

.366295 

8.05 

47 

.252038 

6.72 

.337578 

8.18 

47 

.275924 

6.52 

.366778 

8.03 

48 

.232491 

6.72 

.3:38069 

8.18 

48 

.276315 

6.52 

.367260 

8.03 

49 

.252394 

6.72 

.338560 

8.17 

49 

.276706 

6.52 

.367742 

8.03 

50 

.253297 

6.70 

.339050 

8.18 

50 

.277097 

6.52 

.368224 

8.03 

51 

9.253699 

6.72 

9.339541 

8.17 

51 

9.277488 

6.50 

9.368706 

8.03 

52 

.254102 

6.70 

.340031 

8.18 

52 

.277878 

6.50 

.369188 

8.03 

53 

.2545)4 

6.70 

.340522 

8.17 

53 

.278268 

6.50 

.369670 

8.02 

51 

.2549;)iJ 

6.70 

.341012 

8.17 

54 

.278658 

6.50 

.370151 

8.05 

55 

.255303 

6.63 

.341502 

8.15  ! 

55 

.279048 

6.50 

.370632 

8  03 

56 

.255709 

6.70 

.341991 

8.17 

5b 

.2794:38 

6.48 

.371114 

8.02 

57 

.230111 

6.68 

.342481 

8.17  | 

57 

.279827 

6.50 

.371595 

8.02 

58 

.256512 

6.68 

.342971 

8.15  1 

58 

.280217 

6.48 

.372076 

8.00 

59 

.256913 

6.68 

.343460 

8.15 

59 

.280606 

6.48 

.372556 

8.02 

60 

9.257314 

6.67 

9.343949 

8.15  i 

60 

9.280995 

6.47 

9.373037 

8.02 

TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


1 

36° 

37' 

> 

/ 

Vers. 

D.  1. 

Ex.  sec. 

D.  I'. 

' 

Vers. 

D.I'. 

Ex.  sec. 

D.  r. 

0 

9.280995 

6.47 

9.373037 

8.02 

j  0 

9.303983 

6.28 

9.401634 

.88 

1 

.281383 

6.48 

.373518 

8.00 

1 

.304360 

6.30 

.402107 

.88 

.281772 

6.47 

.373998 

8.00 

1  2 

.304738 

6.28 

.402580 

.87 

3 

.283160 

6.47 

.374478 

8.00 

I 

.305115 

6.28 

.403052 

.87 

4 

.282548 

6.47 

.374958 

8.00 

|  4 

.305492 

6.27 

j  .403524 

.88 

5 

.282936 

6.47 

.375438 

8.00 

5 

.305868 

6.28 

:  .403997 

.88 

6 

.283324 

6.47 

.375918 

8.00 

6 

.306245 

6.27 

.404469 

.87 

7 

.283712 

6.45 

.376398 

7.98 

7 

.306621 

6.28 

.404941 

.87 

8 

.284099 

6.45 

.37(5877 

8.00 

1  8 

.306998 

6.27 

I  .405412 

.87 

9 

.284486 

6.45 

.377357 

7.98 

Q 

.307374 

6.25 

.405884 

.87 

10 

.284873 

6.45 

.377836 

7.98 

10 

.307749 

6.27 

.406356 

.85 

11 

9.285260 

G.45 

9.378315 

7.  -98 

11 

9.308125 

6.27 

9.406827 

.85 

12 

.285647 

6.43 

.378794 

7.98 

12 

.308501 

6.25 

.407298 

.87 

13 

.286033 

6.43 

.379273 

7.  -98 

13 

.308876 

6.25 

!  .407770 

.85 

14 

.286419 

6.43 

.379752 

7.98 

14 

.309251 

6.25 

.408241 

.85 

15 

.286805 

(5.43 

.380231 

7.97 

15 

.309626 

6.25 

.408712 

.85 

16 

.287191 

6.43 

.380709 

7.98 

16 

.310001 

6.^3 

.409183 

.83 

17 

.287577 

6.42 

.381188 

7.97 

17 

.310375 

6.25 

.409653 

.85 

18 

.287962 

6.40 

.381666 

7.97 

18 

.310750 

6.23 

.410124 

.83 

19 

.288348 

6.42 

.382144 

7.97 

19 

.311124 

6.23 

.410594 

.85 

20 

.288733 

6.42 

.382622 

7.97 

20 

.311498 

6.23 

.411065 

.83 

21 

9.289118 

6.40 

9.383100 

7.95 

21 

9.311872 

6.22 

i  9.411535 

.83 

2-2 

.289502 

6.42 

.383577 

7.97 

28 

.312245 

6.23 

.412005 

.83 

23 

.289887 

6.40 

.384055 

7.95 

23 

.312619 

6.22 

.412475 

.83 

24 

.290271 

6.40 

.384532 

7.97 

24 

.312992 

6.22 

.412945 

.83 

25 

.290655 

6.40 

.385010 

7.95 

25 

.31.3365 

6.22 

.413415 

.8£ 

26 

.291039 

6.40 

.385487 

7.95 

26 

.313738 

6.22 

.413884 

.83 

27 

.291423 

6.40 

.385964 

7.95 

27 

.314111 

0.22 

.414:354 

.82 

28 

.291807 

6.38 

.386441 

7.95 

28 

.314484 

6.20 

.414823 

.83 

29 

.292190 

6.38 

.386918 

7.93 

29 

.314856 

6.20 

.415293 

.82 

30 

.292573 

6.38 

.387394 

7.95 

30 

.315228 

6.20 

.415762 

.82 

31 

9.292956 

6.38 

9.387871 

7.93 

31 

9.315600 

6.20 

9.416231 

.82 

32 

.293339 

6.38 

.388347 

7.95 

32 

.315972 

6.20 

.416700 

.80 

33 

.293722 

6.37 

.388824 

7.93 

33 

.316344 

6.20 

.417168 

.82 

34 

.294104 

6.37 

.389300 

7.93 

34 

.316716 

6.18 

.417637 

.82 

35 

.294486 

6.37 

.389776 

7.93 

35 

.317087 

6.18 

.418106 

.80 

3G 

.294868 

6.37 

.390252 

7.92 

36 

.317458 

6.18 

.418574 

.80 

37 

.295250 

6.37 

.390727 

7.93 

37 

.317829 

6.18 

.419042 

.82 

38 

.295632 

6.37 

.391203 

7.92 

38 

.318200 

6.18 

.419511 

.80 

39 

.296014 

6.35 

.391678 

7.93 

39 

.318571 

6.17 

.419979 

.80 

40 

.296395 

6.35 

.392154 

7.92 

40 

.318941 

6.17 

.420447 

.80 

41 

9.296776 

6.35 

9.392629 

7.92 

41 

9.319311 

6.18 

9.420915 

.78 

42 

.297157 

6.35 

.393104 

7.92 

42 

.319682 

6.15 

.421382 

.80 

43 

.297o38 

6.33 

.393579 

7.92 

43 

.320051 

6.17 

.42ia50 

.78 

44 

.297918 

6.35 

.394054 

7.92 

44 

.320421 

6.17 

.422317 

.80 

45 

.298299 

6.33 

.394529 

7.90 

45 

.320791 

6.15 

.422785 

.78 

46 

.298679 

6.33 

.395003 

7.92 

40 

.321160 

6.17 

.423252 

.78 

47 

.299059 

6.33 

.395478 

7.90 

47 

.321530 

6.15 

.423719 

.78 

48 

.299439 

6.33 

.395952 

7.90 

i  48 

.821899 

6.13 

.424186 

.78 

49 

.299819 

6.32 

.396426 

7.90 

49 

.322267 

6.15 

.424653 

.78 

50 

.300198 

6.32 

.396900 

7.90 

50 

.322636 

6.15 

.425120 

.78 

51 

9-300577 

6.33 

0.397374 

7.90 

ni 

9.323005 

6.13 

9.425587 

.77 

52 

.300957 

6.30 

.397848 

7.!>0 

:>:> 

.3*3373 

6.13 

.4215053 

.78 

53 

.301835 

6.32 

.398322 

7.88 

53 

.32)741 

fi.13 

.426520 

54 

.301714 

6.32 

.398795 

7.90 

54 

.324109 

6.13 

.426986 

'  '.7" 

55 

.302093 

6.30 

.399269 

7.88  | 

55 

.324477 

6.13 

.427452 

p'.7~ 

5G 

.302471 

6.30 

.399742 

7.88 

66 

.324845 

6.12 

.427918 

7  7" 

57 

.302849 

6.30 

.400215 

7.88 

57 

.325212 

6.13 

.428384 

7> 

58 

.303227 

6.30 

.400688 

7.88 

58  i 

.325580 

6.12 

.428850 

7.7" 

59 

.303605 

6.30 

.401161 

7.88 

59  ! 

.325047 

6.12 

.429316 

r~  i~r» 

60 

9.303983 

6.28 

9.401634 

7.88 

60 

9.326314 

6.12 

9.429782 

7!  75 

422 


AND  EXTERNAL  SECANTS. 


38° 

39° 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  r. 

' 

Vers.  ID.  1". 

Ex.  sec.  D.  1". 

I 

0  9.326314 

6.12 

9.429782   7.75 

0 

9.348021 

5.93 

9.457518  i  7.65 

j 

.326(381 

6.10 

.430247 

7.77 

1 

.348377 

5.95 

.457977   7.65 

2 

.327047 

6.12 

.430713 

7.75 

2 

.348734 

5.93 

.458436   7.65 

3 

.327414 

6.10 

.431178   7.75 

3 

.349090 

5.93 

.458895   7.63 

.327780  i  6.10 

.431643 

7.75 

4 

.349446 

5.93 

.459:353 

7.65 

K 

.328146   6.10 

.432108   7.75 

5 

.349802 

5.93 

.459812 

7.63 

6 

.328512   6.10 

.432573  1  7.75 

6 

.850158 

5.93 

.460270   7.65 

7 

.328878   6.08 

.433038 

7.75 

7   .350514 

5.92 

.460729   7.63 

8 

.329243   6.10 

.433503 

7.73 

8 

.350869 

5.93 

.461187   7.63 

9 

.329609   6.08 

.433967 

7.75 

9 

.351225 

5.92 

.461645 

7.63 

10 

.329974 

6.08 

.434432 

7.73 

10 

.351580 

6.92 

.462103 

7.63 

11 

9.330339 

6.08 

9.434896 

7.75 

11 

9.351935 

6.92 

9.462561 

7.63 

12   .330704   6.08 

.435361 

7.73 

12 

.352290 

5.90 

.463019 

7.63 

13   .331069   6.07 

.435825 

7.73 

13 

.352644 

5.92 

.463477 

7.62 

14 

.3314.33  i  6.08 

.436289 

7.73 

14 

..352999 

5.90 

.463934 

7.63 

15 

.331798   6.07 

.436753 

7.73 

15 

.353353 

5.90 

.464392 

7.62 

16 

.332162   6.07 

.437217 

7.72 

16 

.353707 

5.92 

.464849 

7.63 

17   .332526   6.07 

.437680 

7.73 

17 

.354062 

5.88 

.465307 

7.62 

18  '  .332890   6.07 

.438144 

7.73 

18 

.354415 

5.90 

.465764 

7.62 

19 

.333254 

6.05 

.438608 

7.72 

19 

.354769 

5.90 

.466221 

7.62 

20 

.333617 

6.07 

.439071   7.72 

20 

.355123 

5.88 

.466678 

7.62 

21 

9.333981 

6.05 

9.439534 

21 

9.355476 

5.88 

9.467135 

7.62 

22 

.334344  !  6.05 

.439997 

7  72 

22 

.855829 

5.88 

.467592 

7.62 

23 

.334707 

6.05 

.440460 

7^72 

23 

.356182 

5.88 

.468049 

7.62 

24 

.335070 

6.03 

.440923 

7.72 

24 

.356535 

5.88 

.468506 

7.60 

25 

.335432 

6.05 

.441386 

7.72 

25 

.356888 

5.88 

.468962 

7.60 

26 

.335795   6.03 

.441849 

7.72 

26 

.357241 

5.87 

.469418 

7.62 

27 

.336157   6.03 

.442312 

7.70 

27 

.357593 

5.87 

.469875 

7.60 

28 

.336519 

6.03 

.442774 

7.72 

28 

.357945 

5.87 

.470331 

7.60 

29 

.336881   6.03 

.443237 

7.70 

29 

.358297 

5.87 

.470787 

7.60 

30 

.337243   6.03 

.443699 

7.70 

30 

.358649 

5.87 

.471243 

7.60 

31 

9.337605 

6.02 

9.444161 

7.70 

31 

9.359001 

5.87 

9.471699 

7.60 

32 

.337966 

6.03 

.444623 

7.70 

32 

.359353 

5.85 

.472155 

7.60 

33 

.338328 

6.02 

.445085 

7.70 

33 

.359704  6.87 

.472611 

7.60 

34 

.338689 

6.02 

.445547 

7.70 

34 

.860056  5.85 

.473067 

7.58 

35 

.339050 

6.02 

.446009 

7.68 

35 

.360407 

5.85 

.473522. 

7.eo 

36 

.339411 

6.00 

.446470 

7.70 

1  36 

.360758 

5.83 

.473978 

7.58 

37 

.339771 

6.02 

.446932 

7.68 

37 

.861108 

5.85 

.474433 

7.58 

38 

.340132 

6.00 

.447393 

7.70 

38 

.361459 

5.85 

.474888 

7.58 

39 

.340492 

6.00 

.447855 

7.68 

89 

.361810 

5.83 

.475343 

7.58 

40 

340852 

6.00 

.448316 

7.68 

40 

.362160 

5.83 

.475798 

7.58 

41 

9.341212 

6.00 

9.448777 

7.68 

41 

9.S62510 

5.83 

9.476253 

7.58 

42 

.341572 

6.00 

.449238 

7.68 

42 

.362860 

5.83 

.476708 

7.58 

43 

.341932 

5.98 

.449699 

7.68 

43 

.363210 

5.83 

.477163 

7.58 

44 

.342291 

6.00 

.450160 

7.67 

44 

.363560 

5.82 

.477618 

7.57 

45 

.342651 

5.98 

.450620 

7  68 

45 

.363909 

5.83 

.478072   7.58 

46 

.343010 

5.98   .451081 

7.67 

46 

.364259 

5.82 

.478527 

7.57 

47 

.343369 

5.98 

.451541 

7.68 

47 

.364608 

5.82 

.478981 

7.57 

48 

.343728 

5.97 

.452002   7.67 

48 

.364957  5.82 

.479435 

7.58 

49 

.344086 

5.98 

.452462 

7.67 

49 

.365306 

5.82 

.479890   7.57 

50 

.344445 

5.97 

.452922 

7.67 

50 

.365655 

5.80 

.480344 

7.57 

51 

9.344803 

5.97 

9.453382 

7.67 

61 

9.366003 

5.82 

9.480798 

7.57 

52 

.345161 

5.97 

.453842 

7.67 

52 

.366352 

5.80 

.481252 

7.55 

53 

.345519 

5.97 

.454302 

7.67 

53 

.366700 

5.80 

.481705 

7.57 

54 

.345877 

5.97 

.454762   7.65 

54 

.367048 

5.80 

.482159  1  7.57 

55 

.346235 

5.95 

.455221   7.67 

55 

.367396 

5.80   .482613 

7.55 

56 

.346592 

5.97 

.455681   7.65 

56 

.367744 

5.78   .483066 

7.57 

57 

.346950 

5.95 

.456140 

7.67 

57  H  .368091 

5.80 

.48:3520 

7.55 

58 

.347:307 

5.95 

.456600 

7.65 

58  1  .368439 

5.78 

.483973 

7.55 

59 

.347664 

5.95 

.457059 

7.65 

59 

.368786 

5.78 

.484426 

7.55 

60 

9.348021 

5.93 

9.457518 

7.65 

60 

9  369133 

5.78 

9.484879   7.55 

423 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


40° 

41° 

/ 

Vers. 

D.  r. 

Ex.  sec.  D.  1". 

' 

Vers.   D.  1". 

Ex.  sec. 

D.  1". 

0 

9.369133 

5.78 

9.484879   7.55 

0  i  9.389681  5.62 

9.511901   7.42 

1 

.369480 

5.78 

.485332 

7.55 

1 

.390018  i  5.63 

.512348 

7.47 

2 

.369827   5.78 

.485785 

7.55 

2 

.390a56  5.63 

.512796 

7  45 

3 

.370174 

5.77 

.486238 

7.55 

3 

.390094  |  5.62 

.513243 

7.47 

4 

.370520   5.78 

.486691 

7.55 

4 

.391031   5.62 

.513091 

7.45 

5 

.370867  i  5.77 

.487144 

7.53 

5 

.391368  5.62 

.514138 

7.45 

G 

.371213   5.77  i  .487596 

7.55 

6 

.391705 

5.62 

.514585 

7.47 

7   .371559   5.77   .488049 

7.53 

7 

.392042 

5.62 

.515033 

7.45 

8   .371905   5.77   .488501 

7.53 

8 

.392379 

5.02 

.515480 

7.45 

9   .372351   5.75 

.488953 

7.55 

9 

.392716 

5.60 

.515927 

7  45 

10  !  .372598  1  5.77 

.489406 

7.53 

10 

.393052 

5.60 

.516374 

7.43 

11  !  9.372942 

5.75 

9.489858 

7.53 

11 

9.393388 

5.60 

9.516820 

7  45 

12 

.373287 

5.75    .490310 

7.53 

12 

.393724 

5.62 

.517267 

7.45 

13 

.373632 

5.75 

.490762 

7.53 

13 

.394061 

5.58 

.517714 

7.43 

14 

.373977 

5.75 

.491214 

7.52 

14 

.394396 

5.00 

.518100 

7.45 

15 

.374322 

5  .  75 

.491665 

7.53 

15 

.394732 

5.60 

.518007 

7.43 

10 

.374667 

5.73 

.492117 

7.53 

16 

.395068 

5.58 

.519053 

7.45 

17 

.375011 

5.75 

.492569 

7.52 

17 

.395403 

5.58 

.519500 

7  43 

18 

.375356 

5.73 

.493020 

7.52 

18 

.895788 

5.60 

.519946 

7.43 

19   .375700 

5.73 

.493471 

7.53 

19 

.396074 

5.58 

.520392 

7.43 

20 

.376044 

5.73 

.493923 

7.52 

20 

.396409 

5.57 

.520838 

7.43 

21 

9.376388 

5.73 

9.494374 

7.52 

21 

9.396743 

5.58 

9.521284 

7.43 

22 

.376732 

5.72 

.494825 

7.52 

21 

.397078 

5.58 

.521730 

7  43 

23 

.377075 

5.73 

.495276 

7.52 

23 

.397413 

5.57 

.522176 

7.42 

24 

.377419 

5.72 

.495727 

7.52 

24 

.397747 

5.57 

.522621 

7  43 

25 

.377762 

5.72 

.496178 

7.50 

25 

.398081 

5.57 

.523067 

7.43 

23 

.378105 

5.72 

.496028 

7.52 

26 

.398415 

5.57 

.523513 

7  42 

27 

.378448 

5.72 

.497079 

7.52 

27 

.398749 

5.57 

.523958 

7.43 

23 

.378791 

5.70 

.497530 

7.50 

28 

.3990a3 

5.57 

.524404 

7  42 

23 

.379133 

5.72 

.497980 

7.52 

29 

.399417 

5.55 

.524849 

7.42 

30 

.379476 

5.70 

.498430   7.50 

30 

.399750 

5.57 

.525294 

7.42 

31 

9.379818 

5.72 

9.498881 

7.48 

31 

9.400084 

5.55 

9.525?39 

7.42 

32 

.380161 

5.70   .499331 

7.52 

32 

.400417 

5.55 

.520184 

7.42 

33 

.389503 

5.70 

.499781 

7.50 

33 

.400750 

5.55 

.526629 

7.42 

34 

.380845 

5.68 

.500231 

7.50 

34 

.401083 

5.55 

.527074 

7.42 

35 

.381186 

5.70 

.500681 

7.50 

&5 

.401416 

5.53 

.527519 

7  42 

3(3 

.381528 

5.68 

.501131 

7.50 

36 

.4017'48 

5.55 

.527964 

7.42 

37 

.381869 

5.70 

.501581 

7.48 

37 

.402081 

6.53 

.528409 

7  40 

38 

.882211 

5.68 

.502030 

7.50 

38 

.402413 

5.53 

.528853 

7.42 

39 

.382552 

5.68 

.502480 

7.48 

39 

.402745 

5.53 

.529298 

7  40 

40 

.382893 

6.68 

.502929 

7.50 

40 

.403077 

5.53 

.529742 

7.42 

41 

9.383234 

5.67 

9.503379 

7.48 

41 

9.403409 

5.53 

9.530187 

7  40 

42 

.383574 

5.68 

.503828 

7.48 

42 

.403741 

5.53 

.530031 

7.40 

43 

.383915 

5.67 

.504277 

7.48 

43 

.404073 

5.52 

.531075 

7.40 

44 

.384255 

5.67 

.504726 

7.48 

44 

.404104 

5.53 

.531519 

7.40 

45 

.384595 

5.67 

.505175 

7.48 

45 

.404736 

5.52 

.531903 

7.40 

46 

.384935 

5.67 

.505024 

7.48 

46 

.405067 

5.52 

.532407 

7.40 

47 

.385275 

5.67 

.506073 

7.48 

47 

.405398 

5.52 

.532851 

7  40 

48 

.385615 

5.67 

.506522 

7.48 

48 

.405729 

5.50 

.533295 

7.40 

49 

.385955 

5.65 

.506971 

7.47 

49 

.406059 

5.52 

.533789 

7.38 

50 

.386294 

5.67 

.507419 

7.48  | 

50 

.406390 

5.52 

.534182 

7.40 

51 

9.386634 

5.65 

9.507868 

7.47 

51 

9.406721 

5.50 

9.534628 

7.40 

52 

.386973 

5.65 

.508316 

7.48   52 

.407051 

5.50 

.636070 

7.38 

53 

.887312 

5.65 

.508765 

7.47 

53 

.407381 

5.50 

.535513 

7.38 

54 

.387651 

5.63 

.509213 

7.47 

54 

.407711 

5.50 

.535956 

7.40 

55 

.387989 

5.65 

.509661 

7.47 

55 

.408041 

5.50 

.530400 

7.38 

56 

.388328 

5.63 

.510109 

7.47 

56 

.408371 

5.48 

.530843 

7.38 

57 

.3386(56 

5.65 

.510557 

.  7.47 

57 

.408700 

5.50 

.537280 

7.38 

58 

.389005 

5.63 

.511005 

7.47 

58 

.409030 

5.48 

.537729 

7.38 

59 

.389343 

5.63 

.511453 

7.47 

59 

.409359 

5.48 

.533172 

7.38 

60 

9.389681 

5.62 

9.511901 

7.45 

60  i  9.409088 

5.48 

9.538015 

7.38 

424 


AND  EXTERNAL  SECANTS. 


42° 

'  "I 
43° 

/ 

Vers. 

D.I'. 

Ex.  sec. 

D.  r. 

/ 

Vers. 

D.  r. 

Ex.  sec.  D.  1". 

0   0.409688 

5.48 

9.538615 

7.38 

0  9.429181 

5.35 

9.565053 

7.32 

I 

.410017 

5.48 

.539058 

7.37 

1 

.429502 

5.33 

.565492 

7.30 

2 

.410346 

5.48 

.539500 

7.38 

2 

.429822 

5.a3 

.565930 

7.32 

3 

.410675 

5.48 

.539943 

7.38 

3 

.430142 

5.35 

.566369 

7.30 

4 

.411004 

5.47 

.540386 

7.37 

4 

.430463 

5.33 

.566807 

7.30 

5 

.411332 

5.47 

.540828 

7.38 

5 

.430783 

5.a3 

.567245 

7.30 

6 

.411600 

5.48 

.541271 

7.37 

6 

.431103 

5.32 

.567683 

7.30 

7 

.411989 

5.47 

.541713 

7.37 

7 

.431422  5.33 

.568121 

7.30 

8 

.412317 

5.45 

.542155 

7.37 

8 

.431742 

5.33 

.568559 

7.30 

9 

.412644 

5.47 

.542597 

7.38 

9 

.432062 

5.32 

.568997 

7.30 

10 

.412972 

5.47 

.543040 

7.37 

10 

.432381 

5.32 

.569435 

7.30 

11 

9.413300* 

5.45 

9.543482 

7.37 

11 

9.432700 

5.33 

9.569873 

7.30 

13 

.413627 

5.47 

.543924 

7.37 

12 

.4830-20 

5.32 

.570311 

7.28 

13 

.413955 

5.45 

.544366 

7.35 

13 

.4a3339 

5.30 

.570748 

7.30 

14 

.414282 

5.45 

.544807 

7.37 

14 

.433657 

5.32 

.571186 

7.30 

15 

.414609 

5.45 

.545249 

7.37 

15 

.433976 

5  32 

.571624 

7.28 

16 

.414936 

5.45 

.545691 

7.a5 

16 

.434295 

5.30 

.572061 

7.28 

17 

.415263 

5.43 

.546132 

7.37 

17 

.434613 

5.32 

.572498 

7.30 

18 

.413389 

5.45 

.546574 

7..  35 

18 

.434932 

5.30 

.572936 

7.28 

19 

.415916 

5.43 

.547015 

7.37 

19 

.4a5250 

5.30 

.573373 

7.28 

20 

.416242 

5.43 

.547457 

7.35 

20 

.435568 

5.30 

.573810 

7.28 

21 

9.416568 

5.43 

9.547898 

7.a5 

21 

9.435886 

5.30 

9.574247 

7.30 

22 

.41(5894 

5.43 

.548339 

7  37 

22 

.436204 

5.28 

.574685 

7.28 

23 

.417220 

5.43 

.548781 

7.35 

23 

.436521 

5.30 

.575122 

7.27 

24 

.417546 

5.42 

.549222 

7.35 

24 

.436839 

5.28 

.575558 

7.30 

25 

.417871 

5.43 

.549663 

7.35 

25 

.437156 

5.28 

.575995 

7.28 

26 

.418197 

5.42 

.550104 

7.33 

26 

.437473 

5.30 

.576432 

7.28 

27 

.418522 

5.43 

.550544 

7.35 

27 

.437791 

5.27 

.576869 

7.28 

28 

.418848 

5.42 

.550085 

7  '85 

28 

.438107 

5.28 

.577306 

7.27 

29 

.419173 

5.42 

.551426 

7.35 

29 

.438424 

5.28 

.577742 

7.28- 

30 

.419498 

5.40 

.551867 

7.33 

30 

.438741 

5.28 

.578179 

7.27 

31 

9.419822 

5.42 

9.552307 

7.35 

31 

9.439058 

5.27 

9.578615 

7.28 

32 

.420147 

5.40 

.552748 

7.33 

32 

.439374 

5.27 

.579052 

7.27' 

.33 

.420171 

5.42 

.553188 

7.35 

33 

.439690 

5.28 

.579488 

7.27 

34 

.420796 

5.40 

.553629 

7.33 

34 

.440007 

5.27 

.579924 

7.28 

35 

.421120 

5.40 

.554069 

7.33 

35 

.440:323 

5.27 

.580361 

7.27 

36 

.421414 

5.40 

.554509 

7.33 

36 

.440639 

5.25 

.580797 

7.27 

37 

.421768 

5.40 

.554949 

7.33 

37 

.440954 

6.27 

.581233 

7.27. 

as 

.422092 

5.40 

.555389 

7.33 

38 

.441270 

5.25 

.581609 

7  27 

39 

.422416 

5.38 

.555829 

7.33 

39 

.441585 

5.27 

.582105 

7^27 

40 

.422739 

5.40 

.556269 

7.33 

40 

.441901 

5.25 

.582541 

7.27 

41 

9.423063 

5.38 

9.556709 

7.33 

41 

9.442216 

5.25 

9.582977 

7.27 

42 

.423386 

5.38 

.557149 

7.33 

42 

.442531 

5.25 

.583413 

7.25 

43 

.423709 

5.38 

.557c89 

7.32 

43 

.442846 

5.25 

.583848 

7.27 

41 

.424032 

5.38 

.558028 

7.33 

44 

.443161 

5.28 

.584284 

7.27 

45 

.424355 

5.37 

.558468 

7.32 

45 

.443476 

5.23 

.584720 

7.25 

46 

.424677 

5.38 

.558907 

7.as 

46 

.443790 

5.25 

.585155 

7.27 

47 

.425000 

5.37 

.559347 

7.32 

47 

.444105 

5.23 

.585591 

7.25 

48 

.425322 

5.38 

.559786 

7.33 

48 

.444119 

5.23 

.586026 

7.27 

49 

.425645 

5.37 

.560226 

7.32 

49 

.444733 

5.23 

.586462 

7.25 

50 

.425967 

5.37 

.560665 

7.  S3 

50 

.445047 

5.23 

.586897 

7.25 

51 

9.426289 

5.37 

9.561104 

7.32 

51 

9.445361 

5.23 

9.587332 

7.25 

52 

.426611 

5.37 

.561543 

7.32 

52 

.445675 

5.23 

.587767 

7.27 

53 

.426933 

5.35 

.561982 

7.32 

53 

.445989 

5.22 

.588203 

7.25 

51 

.427254 

5.37 

.562421 

7.32 

54 

.446302 

5.23 

.588638 

7.25 

55 

.427576 

5.35 

.562860 

7.32 

55 

.446616 

5.22 

.589073 

7.25 

56 

.427897 

5.35 

.563299 

7.32 

56 

.446929 

5.22 

.589508 

7.23 

57 

.428218 

5.35 

.563738 

7.30 

57 

.447242 

5.22 

.589942 

7.25 

58 

.428539 

5.35 

.564176 

7.32 

58 

.447555 

5.22 

.590377 

7.25 

59 

.428860 

5.35 

.564615 

7.30 

59 

.447868 

5.22 

.590812 

7.25 

6) 

9.429181 

5.33 

9.565053 

7.32 

60 

9.448181 

5.20 

9.591247 

7.23 

435 


TABLE  XXVI. -LOGARITHMIC  VERSED  SINES 


44° 

45° 

/ 

Vers. 

D.  r. 

Ex.  sec. 

D.  1". 

' 

Vers. 

D.I-. 

Ex.  sec. 

D.  r. 

0 

9.448181 

5.20 

9.591247 

7.23 

0 

9.466709 

5.08 

9.617224 

7.20 

1 

.448493 

5.22 

.591681 

7.25 

1 

.467014 

5.08 

.617656 

7.18 

2 

.448806 

5.20 

.592116 

7.25 

2 

.467319 

5.08 

.618087 

7.18 

3 

.449118 

5.22 

.592551 

7.23 

3 

.467624 

5.07 

.618518 

7.18 

4 

.449431 

5.20 

.592985 

7.23 

4 

.467928 

5.08 

.618949 

".18 

5 

.449743 

5.20 

.593419 

7.25 

5 

.468233 

5.07 

.619380 

7.18 

6 

.450055 

5.18 

.593854 

7.23 

6 

.468537 

5.07 

.619811 

".18 

7 

.450366 

5.20 

.594288 

7.23 

7 

.468841 

5.07 

.620242 

".18 

8 

.450678 

5.20 

.594722 

7.23 

8 

.469145 

5.07 

.620673 

".18 

9 

.450990 

5.18 

.595156 

7.25 

9 

.469449 

5.07 

.621104 

".18 

10 

,451£01 

5.18 

.595591 

7.23 

10 

.469753 

5.07 

.621535 

".18 

11 

9.451612 

5.20 

9.596025 

•7.23 

11 

9.470057 

5.05. 

9.621966 

".17 

12 

.451924 

5.18 

.596459 

7.23 

12 

.470360 

5.07 

.622396 

".18 

13 

.452235 

5.18 

.596893 

7.22 

13 

.470664 

5.05 

.622827 

".18 

14 

.452546 

5.17 

.597326 

7.23 

14 

.470967 

5.05 

.623258 

".17 

15 

.452856 

5.17 

.597760 

7.23 

!  15 

.471270 

5?05 

.623688 

".18 

16 

.453167 

5.18 

.598194 

7.23 

!  16 

.471573 

5.05 

.624119 

".17 

17 

.453478 

5.17 

.598628 

7.22 

17 

.471876 

5.05 

.624549 

".18 

18 

.453788 

5.17 

.599061 

7.23' 

18 

.472179 

5.05 

.624980 

".17 

19 

'  .454098 

5.17 

.599495 

7.22 

19 

.472482 

5.03 

.625410 

".18 

20 

.454408 

5.17 

.599928 

7.23 

1  20 

.472784 

5.05 

.625841 

".17 

21 

9.454718 

5.17 

9.G003G2 

7.22 

21 

9.473087 

5.03 

9.C26271 

".17 

22 

.455028 

5.17 

.600795 

7.23 

22 

.473389 

5.03 

.626701 

".17 

23 

.455338 

5.17 

.601229 

7  22 

23 

.473691 

5.03 

.627131 

".17 

24 

.455648 

5.15 

.601662 

7^22 

24 

.473993 

5.03 

.627561 

7.17 

25 

.455957 

5.17 

.602095 

7.22 

25 

.474295 

5.03 

.627991 

.17 

26 

.456267 

5.15 

.602,328 

7.23 

26 

.474597 

5.03 

.628421 

.17 

27 

.456576 

5.15 

.602062 

7.22 

27 

.474899 

5.02 

.628851 

.17 

28 

.456885 

5.15 

.603395 

7.22 

28 

.475200 

5.03 

.629281 

.17 

29 

.457194 

5.15 

.603828 

7.22  • 

29 

.475502 

5.02 

.629711 

.17 

30 

.457503 

5.13 

.604261 

7.22 

30 

.475803 

5.02 

.630141 

.17 

31 

9.457811 

5.15 

9.604694 

7.20 

31 

9.476104 

5.02 

9.630571 

.17 

32 

.458120 

5.15 

.605126 

7.22  i 

i  32 

.476405 

5.02 

.631001 

.15 

33 

.458429 

5.13 

.605559 

7.22  ! 

i  33 

.476706 

5.02 

.631430 

.17 

34 

.458737 

5.13 

.C05992 

7.22 

34 

.477007 

5.02 

.631860 

.17 

35 

.450045 

5.13 

.606425 

7.20 

35 

.477308 

5.  CO 

.632280 

.15 

36 

.459353 

5.13 

.606857 

7.22  j 

36 

.477608 

5.02 

.632719 

.17 

37 

.459061 

5.13 

.607290 

7.20 

37 

.477909 

5.  CO 

.633149 

.15 

33 

.450969 

5.13 

.607722 

7.2'2 

38 

.478209 

5.  CO 

.633578 

.17 

30 

.460277 

5.12 

.608155 

7.20  ! 

39 

.478509 

5.00 

.634008 

.15 

40 

.460584 

5.13 

.608587 

7.22 

40 

.  .478809 

5.  CO 

.634437 

.15 

41 

9.460S92 

5.12 

9.609020 

7.20 

41 

9.479109 

5.  CO 

9.634866 

.17 

42 

.461199 

5.12 

.609452 

7.20 

42 

.479409 

5.  CO 

.635296 

.15 

43 

.461506 

5.12 

.609884 

7.20 

43 

.4797'09 

5.00 

.635725 

.15 

44 

.461813 

5.12 

.61031(3 

7.22 

44 

.480009 

4  98 

.636154 

.15 

ft 

.462120 

5.12 

.610749 

7.20 

45 

.480308 

5.  CO 

.636583 

.15 

40 

.462427 

5.12 

.611181 

7.20 

46 

.480608 

4.98 

.637012 

.15 

47 

.462734 

5.10 

.611613 

7.20 

47 

.480907 

4.98 

.637441 

.15 

48 

.463040 

5.12 

.612045 

7.20 

48 

.481206 

4.98 

.637870 

.15 

49 

.463347 

5.10 

.612477 

7.18 

49 

.481505 

4.98 

.688299- 

.15 

50 

.463653 

5.10 

.612908 

7.20  i 

50 

.481804 

4.98 

.028728 

.15 

51 

9.463f5D 

5.10 

9.613340 

7.20 

51 

9.482103 

4.97 

9.639157 

.15 

52 

.464265 

5.10 

.613772 

7.20 

52 

.482401 

4.98 

.639586 

.15 

53 

.464571 

5.10 

.614204 

7.18 

53 

.4P2700 

4.97 

.640015 

.13 

54 

.464877 

5.10 

.614635 

7.20  1 

54 

.482988 

4.97 

.640443 

.15 

55 

.465183 

5.08 

.615067 

7.20 

55 

.483296 

4.98 

.64087'2 

.15 

50 

.465488 

5.10 

.615499 

7.18  i 

56 

.483595 

4.97 

.641301 

.13 

57 

.465794 

5.08 

.615930 

7.20 

57 

.483893 

4:97 

.641729 

.15 

58 

.4fi»5099 

5.08 

.616362 

7.18 

58 

.484191 

4.95 

!  642158 

.13 

59 

.466404 

5.08 

.616793 

7.18 

59 

.484488 

4.97 

.642586 

.15 

60 

9.466709 

5.08 

9.617224 

7.20 

60 

9.484786 

4.97 

9.643015 

.13 

426 


AND  EXTERNAL  SECANTS. 


46o                                      470 

/ 

Vers. 

D.  1. 

Ex.  sec. 

D.I". 

/ 

Vers. 

D.I*. 

Ex.  sec. 

D.  1". 

0 

9.484786 

4.97 

9.643015 

7.13 

0 

9.502429 

4.85 

9.668646 

7.10 

1 

.4&5084 

4.95 

.643443 

7.15 

1 

.502720 

4.83 

.669072 

7.10 

2 

.485381 

4.95 

.643872 

7.13 

2 

.503010 

4.83 

.669498 

7.10 

3 

.485678 

4.97 

.644300 

7.13 

3 

.503300 

4.85 

.669924 

7.10 

4 

.485976 

4.95 

.644728 

7.13 

4 

.503591 

4.83 

.670350 

7.10 

5 

.486273 

4.95 

.645156 

7.15 

5 

.503881 

4.83 

.670776 

7.08 

6 

.486570 

4.93 

.645585 

7.13  i 

6 

.504171 

4.82 

.671201 

7.10 

7 

.486866 

4.95 

.646013 

7.13  i 

7 

.504460 

4.83 

.671627 

7.10 

8 

.487163 

4.95 

.646441 

7.13 

8 

.504750 

4.83   .672053 

7.10 

9   .487460 

4.93 

.646869 

7.13 

9 

.505040 

4.82 

.672479 

7.08 

10  |  .487750 

4.95 

.647297 

7.13 

10 

.505329 

4.82 

.672904 

7.10 

11   9.483053 

4.93 

9.647725 

7.13 

11 

9.505618 

4.83 

9.673330 

7.10 

12   .483349   4.93 

.648153 

7.13 

12 

.505908 

4.82 

.673756 

7.08 

13 

.483(345   4.93 

.648581 

7.13 

13 

.506197 

4.82 

.674181 

7.10 

14 

.488941  !  4.93 

.649009 

7.12  i 

14 

.506486 

4.82 

.674607 

7.08 

15 

.489237 

4.93 

.649436 

7.13  i 

15  !  .506775 

4.80 

.675032 

7.10 

16 

.489533 

4.92 

.649864   7.13 

16 

.507063 

4.82 

.675458 

7.08 

17   .489323 

4.93 

.650292   7.13  i  17 

.507352 

4.80   .675883 

7.10 

18   .490124   4.92 

.650720 

7.12  1 

18 

.507640 

4.82  1  .676309 

7.08 

19 

.490119   4.92 

.651147 

7.13  i 

19 

.507929 

4.80 

.676734 

7.08 

23 

.430714 

4.93    .651575 

7.12  !  20 

.508217 

4.80 

.677159 

7.08 

21  9.491010 

4.92  i  9.652002 

7.13  i 

21 

9.508505 

4.80 

9.677584 

7.10 

22   .491305 

4.9:3    .652430 

7.12  ! 

22 

.508793 

4.80 

.678010 

7.08 

23   .491600 

4.93 

.652857 

7.13  i  23 

.509081 

4.80 

.678435 

7.08 

24   .491894 

4.92 

.653285 

7  12 

24 

.509369 

4.80 

.678860 

7.08 

25   .492189 

4.92    .653712 

7^13 

1  25 

.509657 

4.80 

.679285 

7.08 

28   .492484 

4.90 

.654140 

7.12 

26 

.503945 

4.78 

.679710 

7.10 

27   .492778 

4.90 

.654567 

7.12 

27 

.510232 

4.80 

.680136 

7.08 

23   .493072 

4.92    .654994 

7.12 

28 

.510520 

4.78 

.680561 

7.08 

29 

.493367 

4.90    .655421   7.13 

29 

.510807 

4.78 

.680986 

7.08 

30 

.493661 

4.90   .655849  |  7.12 

30 

.511094 

4.78 

.681411 

7.08 

31 

9.493955 

4.90 

9.656276  i  7.12 

31 

9.511381 

4.78 

9.681836 

7.07 

32 

.494249 

4.83 

.656703 

7.12  ! 

32 

.511668 

4.78 

.682260 

7.08 

33 

.494542 

4.90 

.657130 

7.12  i 

33 

.511955 

4.77 

.682685 

7.08 

34 

.494836 

4.90 

.657557 

7.12  ! 

34 

.512241 

4.78 

.683110 

7.08 

35 

.495130 

4.88 

.657984 

7.12  i 

35 

.512528 

4.78 

.683535 

7.08 

36 

.495423 

4.88 

.658411 

7.12  i 

36 

.512815 

4.77 

.683960 

7.08 

37 

.495716 

4.88 

.658833 

7.12  ! 

37 

.513101 

4.77 

.684385 

7.07 

38 

.493009 

4.88 

.659265 

7.10 

38 

.513387 

4.77 

.684809 

7.08 

39 

.493302   4.88 

.659891 

7.12 

39 

.513673 

4.77 

.685234 

7.08 

40 

.498595 

4.88 

.660118 

7.12  | 

40 

.513959 

4.77 

.685659 

7.07 

41 

9.496883 

4.83 

9.660545 

7.12 

41 

9.514245 

4.77 

9.686083 

7.08 

42 

.497181 

4.87 

.660972 

7.10  | 

42 

.514531 

4.77 

.686508 

7.08 

43 

.497473 

4.83 

.661398 

7.12 

43 

.514817 

4.75 

.  .686933 

7.07 

44 

.497766 

4.87 

.661825 

7.12  | 

44 

.515102 

4.77 

.687a57 

7.08 

45 

.493058 

4.87 

.662252 

7.10  ! 

45 

.515388 

4.75 

.687782 

7.07 

46 

.493:350 

4.88 

.662678 

7.12 

46 

.515673   4.77 

.688206 

7.08 

47 

.493643 

4.87 

.683105 

7.10 

47 

.515959 

4.75 

.688631 

7.07 

48 

.493935 

4.S5 

.663531 

7.12 

48 

.516244 

4.75 

.689055 

7.07 

49 

.499896 

4.87 

.663958 

7.10 

49 

.516529 

4.75 

.689479 

7.08 

50 

.499518 

4.87 

.664384 

7.10 

50 

.516814 

4.73 

.689904 

7.07 

51 

9.499810 

4.R5 

9.664810 

7.12 

51 

9.517093 

4.75 

9.690328 

7.07 

52 

.500101 

4.87 

.665237 

7.10 

52 

.517383 

4.75 

.690752 

7.08 

53 

.500393 

4.85 

.665663 

7.10 

53 

.517663 

4.73 

.691177 

7.07 

54 

.500634   4.8'> 

.666080 

7.10 

54 

.517952 

4.73 

.691601 

7.07 

55 

.500975 

4.85 

.666515 

7.12 

55 

.518236 

4.75 

.692025 

7.07 

56 

.501266 

4.85 

.666942 

7.10 

SB 

.518521 

4.73 

.692449 

7.07 

57 

.501557 

4.85 

.667368 

7.10 

57 

.618805 

4.73 

.692873 

7.08 

58 

.501848 

4.85 

.667794 

7.10 

58 

.519089 

4.73 

.693298 

7.07 

59 

.502199 

4.83 

.668220 

7.10 

59 

.519373 

4.73 

.693722 

7.07 

60 

9.502429 

4.85 

9.668646 

7.10 

60 

9.519657 

4.72 

9.694146 

7.07 

427 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


48° 

43° 

Vers. 

D.I". 

Ex.  sec. 

D.  1'. 

' 

Vers. 

D.  r. 

Ex.  sec. 

D.  1". 

0 

9.519657 

4.72 

9.694146 

7.07 

0 

9.536484 

4.62 

9.719541 

7.05 

1 

.519940 

4.73 

.694570 

7.07 

1 

.536761 

4.62 

.719964 

7.03 

2 

.520224 

4.72 

.694994 

7.07 

2 

.537038 

4.62 

.720386 

.05 

3 

.520507 

4.73 

.695418 

7.07 

3 

.537315 

4.62 

.720809 

.03 

4 

.520791 

4.72 

.695842 

7.07 

4 

.537592 

4.62 

.721231 

.03 

5 

.521074 

4.72 

.696266 

7.05  i 

5 

.537869 

4.60 

.721053 

.05 

6 

.521357 

4.72 

.696689 

7.07  i 

G 

.538145 

4.62 

.722076 

.03 

7 

.521040 

4.72 

.697113 

7.07 

7 

.538422 

4.60 

.722498 

.05 

8 

.521923 

4.72 

.697537 

7.07 

8 

.538698 

4.60 

.722921 

.03 

g 

.522206 

4.70 

.697961 

7.07 

9 

.538974 

4.62 

.723343 

.03 

10 

.522488 

4.72 

.698385 

7.07 

10 

.539251 

4.60 

.723765 

.05 

11 

9.522771 

4.72 

9.698809 

7.05 

11 

9.539527 

4.60 

9.724188 

.03 

12 

.523054 

4.70 

.699232 

7.07 

12 

.539803 

.60 

.724010 

.03 

13 

.523336 

4.70 

.699656 

7.07 

13 

.540079 

.58 

.725032 

.03 

14 

.523618 

4.70 

.700080 

7.05 

14 

.540354 

.60 

.725454 

.05 

15 

.523900 

4.70 

.700503 

7.07 

15 

.540630 

.60 

.725877 

.03 

16 

.524182 

4.70 

.700927 

7.05 

16 

.540906 

.58 

.726299 

.03 

17 

.524464 

4.70 

.701350 

7.07 

17 

.541181 

.58 

.7267'21 

.03 

18 

.524746 

4.70 

.701774 

7.07 

18 

.541456 

.60 

.727143 

.03 

19 

.525028 

4.68 

.702198 

7.05 

19 

.541732 

.58 

.727565 

.05 

20 

.525309 

4.70 

.702621 

7.  07 

20 

.542007 

.58 

.727988 

.03 

21 

9.525591 

4.68 

9.703045 

7.05 

21 

9.542282 

4.58 

9.728410 

.03 

22 

.525872 

4.68 

.703468 

7.05 

22 

.542557 

4.58 

.728832 

.03 

23 

.526153 

4.70 

.703891 

7.07 

23 

.542832 

4.57 

.729254 

.03 

24 

.526435 

4.68 

.704315 

7.05 

24 

.543106 

4.58 

.729676 

.03 

25 

.526716 

4.68 

.704738 

7.07 

25 

.543881 

4.57 

.730098 

.03 

26 

.526997 

4.67 

.705162 

7.05 

26 

.543655 

4.58 

.730520 

.03 

27 

.527277 

4.68 

.705585 

7.05 

27 

.543930 

4.57 

.730942 

.03 

28 

.527558 

4.68 

.7060;  8 

7.05 

28 

.544204 

4.57 

.731364 

.03 

29 

.527839 

4.67 

.706431 

7.07 

29 

.544478 

4.57 

.731786 

.03 

30 

.528119 

4.68 

.706855 

7.05 

30 

.544752 

4.57 

.732208 

.03 

31 

9.528400 

4.67 

9.707278 

7.05 

31 

9.545026 

4.57 

9.732630 

.03 

32 

.528680 

4.67 

.707701 

7.05 

32 

.545300 

4.57 

.733052 

.03 

33 

.528960 

4.67 

.708124 

7.05 

33 

.545574 

4.57 

.733474 

.03 

34 

.529240 

4.67 

.708547 

7.07 

34 

.545848 

4.55 

.733896 

.02 

35 

.529520 

4.67 

.708971 

7.05 

35 

.546121 

4.57 

.734317 

.03 

36 

.529800 

4.67 

.709394 

7.05 

i  36 

.546395 

4.55 

.734739 

.03 

37 

.530080 

4.65 

.709817 

7.05 

37 

.540668 

4.55 

.735161 

.03 

38 

.530359 

4.67 

.710240 

7.05 

38 

.546941 

4.55 

.735583 

.03 

39 

.530639 

4.65 

.710663 

7.05 

39 

.547214 

4.55 

.736005 

.03 

40 

.530918 

4.67 

.711086 

7.05 

40 

.547487 

4.55 

.736427 

.02 

41 

9.531198 

4.65 

9.711509 

7.05 

41 

9.547760 

4.55 

9.736848 

.03 

42 

.531477 

4.65 

.711932 

7.05 

42 

.548033 

4.55 

.737270 

.03 

43 

.531756 

4.65 

.712355 

7.05 

43 

.548306 

4.55 

.737692 

.03 

44 

.532035 

4.65 

.712778 

7.03 

44 

.548579 

4.53 

.738114 

.02 

45 

.532314 

4.63 

.713200 

7.05 

4.-) 

.5-18851 

4.55 

.738535 

.08 

46 

.532592 

4.65 

.713623 

7.05 

46 

.549124 

4.53 

.738957 

.03 

47 

.532871 

4.65 

.714046 

7.C5 

47 

.549396 

4.53 

.739379 

.02 

48 

.533150 

4.63 

.714469 

7.05 

48 

.549668 

4.53 

.739800 

.03 

49 

.533428 

4.63 

.714892 

7.05 

49 

.549940 

4.53 

.740222 

.03 

50 

.533706 

4.65 

.715315 

7.03 

50 

.550212 

4.53 

.740644 

.02 

51 

9.533985 

4.63 

9.715737 

7.05 

51 

9.550484 

4.53 

9.741065 

.03 

52 

.5:34203 

4.63 

.716160 

7.05 

52 

.550?56 

4.53 

.741487 

.02 

53 

.534541 

4.63 

.716583 

7.03 

53 

.551028 

4.52 

.741908 

.03 

54 

.534819 

.63 

.717005 

7.05 

54 

.551299 

4.53 

.742330 

.02 

55 

.535097 

.62 

.717428 

7.05 

55 

.551571 

4.52 

.742751 

.03 

56 

.535374 

.63 

.717851 

7.03 

56 

.551842 

4.52 

.743173 

.03 

57 

.535652 

.62 

.718273 

7.05 

57 

.552113 

4.52 

.74X595 

.02 

58 

.535929 

.63 

.718696 

7.03 

58 

.552384 

4.53 

.744016 

.03 

59 

.536207 

.62 

.719118 

7.05 

59 

.552656 

4.52 

.744438 

7.02 

60 

9.536484 

.62 

9.719641 

7.05 

60 

9.552927 

4.50 

9.744859 

7.02 

AND  EXTERNAL  SECANTS. 


50° 

1 

51° 

/ 

Vers. 

D.  1*. 

Ex.  sec. 

D.  1", 

/ 

Vers. 

D  1" 

Ex.  sec. 

D.  1'. 

0 

9.552927 

4.50 

9.744859 

7.02 

0 

9.568999 

4.42 

9.770127 

7.02 

1 

.553197 

4.52 

.745280 

7.03 

1 

.569264 

4.40 

.770548 

7.02 

2 

.553468 

4.52 

.745702 

7.02 

2 

.569528 

4.42 

.770969 

7.00 

3 

.553739 

4.50 

.746123 

7.03 

3 

.569793 

4.40 

.771389 

7.02 

4 

.554009 

4.52 

.746545 

7.02 

4 

.570057 

4.42 

.771810 

7.02 

5 

.554280 

4.50 

.746966 

7.03 

5 

.570322 

4.40 

.772231 

7.02 

6 

.554550 

4.50 

.747388 

7.02 

6 

.570586 

4.40 

.772652 

7.02 

7 

.554820 

4.52 

.747809 

7.02 

7 

.570850 

4.40 

.773073 

7.02 

8 

.555091 

4.50 

.748230 

7.03 

8 

.571114 

4.40 

.773494 

7.00 

9 

.555361 

4.50 

.748652 

7.02 

9 

.571378 

4.40 

.773914 

7.02 

10 

.555631 

4.48 

.749073 

7.02 

10 

.571642 

4.40 

.774335 

7.02 

11 

9.555900 

4.50 

9.749494 

7.03 

11 

9.571906 

4.40 

9.774756 

7.02 

12 

.556170 

4.50 

.749916 

7.02 

12 

.572170 

4.40 

.775177 

7.02 

13 

.556440 

4.48 

.750337 

7.02 

13 

.572434 

4.38 

.775598 

7.00 

14 

.556709 

4.50 

.750758 

Y.03 

14 

.572697 

4.38 

.776018 

7.02 

15 

.556979 

4.48 

.751180 

7.02 

15 

.5?2960 

4.40 

.776439 

7.02 

ia 

.557248 

4.48 

.751601 

7.02 

16 

.573224 

4.38 

.776860 

7.02 

17 

.557517 

4.48 

.752022 

7.02 

17 

.57:3487 

4.38 

.777281 

7.02 

10 

.557786 

4.48 

.752443 

7.03 

18 

.573750 

4.38 

.777702 

7.00 

19 

.558055 

4.48 

.752865 

7.02 

19 

.57'4013 

4.38 

.778122 

7.02 

20 

.558324 

4.48 

.753286 

7.02 

20 

.574276 

4.38 

.778543 

7.02 

21 

9.558393 

4.48 

9.753707 

7.02 

21 

9.574539 

4.38 

9.778964 

7.02 

22 

.558862 

4.48 

.754128 

7.02 

22 

.574802 

4.37 

.779385 

7.  CO 

23 

.559131 

4.47 

.754549 

7.03 

23 

.575064 

4.38 

.779805 

7.02 

24 

.559399 

4.47 

.754971 

7.C2 

24 

.575327 

4.37 

.780226 

7.02 

25 

.559667 

4.48 

.755392 

7.02 

25 

.575589 

4.38 

.780647 

7.02 

20 

.559936 

4.47 

.755813 

7.02 

26 

.575852 

4.37 

.781068 

7.00 

27 

.560204 

4.47 

.756234 

7.02 

27 

.576114 

4.37 

.781488 

7.02 

28 

.560472 

4.47 

.756655 

7.02 

28 

.576376 

4.37 

.781909 

7.02 

29 

.560740 

4.47 

.757076 

7.03 

29 

.576638 

4.37 

.782330 

7.02 

30 

.561008 

4.47 

.757498 

7.02 

30 

.576900 

4.37 

.782751 

7.00 

31 

9.561276 

4.47 

9.757919 

7.02 

31 

9.577162 

4.37 

9.783171 

7.02 

32 

.5C1544 

4.45 

.758340 

7.02 

32 

.577424 

4.35 

.783592 

7.02 

33 

.561811 

4.47 

.758701 

7.02 

33 

.577685 

4.37 

.784013 

7.00 

34 

.562079 

4.45 

.759182 

7.02 

34 

.577947 

4.35 

.784433 

7.02 

35 

.562346 

4.45 

.759603 

7.02 

35 

.578208 

4.37 

.784854 

7.02 

36 

.562613 

4.47 

.760024 

7.02 

36 

.578470 

4.35 

.785275 

7.02 

37 

.562881 

4.45 

.760445 

7.02 

37 

.578731 

4.35 

.785696 

7.00 

38 

.563148 

4.45 

.760866 

7.02 

38 

.578992 

4.35 

.786116 

7.02 

39 

.563415 

4.45 

.761287 

7.02 

39 

.579253 

4.35 

.786537 

7.02 

40 

.563682 

4.43 

.761708 

7.02 

40 

.579514 

4.35 

.786958 

7.00 

41 

9.563948 

4.45 

9.762129 

7.02 

41 

9.579775 

4.35 

9.787378 

7.02 

42 

.564215 

4.45 

.762550 

7.02 

42 

.580036 

4.35 

.787799 

7.02 

43 

.564482 

4.43 

.762971 

7.02 

43 

.580297 

4.33 

-.788220 

7.02 

44 

.564748 

4.45 

.763392 

7.02 

44 

.580557 

4.35 

.788641 

7.00 

45 

.565015 

4.43 

.763813 

7.02 

45 

.580818 

.33 

.789061 

7.02 

46 

.565281 

4.43 

.764234 

7.02 

46 

.581078 

.35 

.789482 

7.02 

47 

.565547 

4.43 

.764655 

7.02 

47 

.581339 

.33 

.789903 

7.00 

48 

.565813 

4.43 

.765076 

7.02 

48 

.581599 

.33 

.790323 

7.02 

49 

.566079 

4.43 

.765497 

7.02 

49 

.581859 

.33 

.790744 

7.02 

50 

.560345 

4.43 

.765918 

7.02 

50 

.582119 

.33 

.791165 

7.02 

51 

9.566611 

4.43 

9.766339 

7.02 

51 

9.582379 

4.33 

9.7915S6 

7.00 

52 

.566877 

4.42 

.  7*36760 

7.02 

52 

.582639 

4.32 

.792006 

7.02 

53 

.567142 

4.43 

.767181 

7.02 

i  53 

.582898 

4.33 

.792427 

7.02 

54 

.567408 

4.42 

.767602 

7.00 

54 

.583158 

4.33 

.792848 

7.  CO 

55 

.567673 

4.42 

.768022 

7.02 

55 

.583-118 

4.32 

.793268 

7.02 

56 

.567938 

4.43 

.768443 

7.02 

56 

.583677 

4.32 

.793G89 

7.02 

57 

.5(iS204 

4.42 

.768864 

7.02 

57 

.583936 

4.33 

.794110 

7.02 

58 

.568169 

4.42 

.769285 

7.02 

58 

.584196 

4!32 

.794531 

7.00 

59 

.568734 

4.42 

.769706 

7.02 

59 

.584455 

4.32 

.794951 

7.02 

GO 

9.568999 

4.42 

9.77-0127 

7.02 

60 

9.584714 

4.32 

9.795372 

7.02 

TABLE  XXVI.-LOGARITHMIC  VERSED  SINES 


52° 

63° 

/ 

Vers. 

D.  r. 

Ex.  sec. 

P.r. 

/ 

1 

Vers. 

D.  1". 

Ex.  sec. 

D.I'. 

0 

9.584714 

4.32 

9.795372 

7.02 

0 

9.600083 

4.22 

9.820622 

7.02 

i 

.584973 

4.32 

.795793 

7.00 

1 

.600338 

4.22 

.821043 

7.02 

2 

.585232 

4.32 

.796213 

7.02 

2 

.600591 

4.23 

.821464 

7.02 

3 

.585491 

4.30 

.796634 

7.02 

3 

.600845 

4.22 

.821885 

7.02 

4 

.585749 

4.32 

.797055 

7.02 

4 

.601098 

4.22 

.822306 

7.02 

5 

.586008 

4.30 

.797476 

7.00 

5 

.601351 

4.20 

.822727 

7.02 

6 

.586266 

4.32 

.797896 

7.02 

6 

.601603 

4.22 

.823148 

7.02 

7 

.5E658) 

4.30 

.798317 

7.02 

7 

.601856 

4.22 

.823569 

7.02 

8 

.58678) 

4.30 

.798738 

7.00 

8 

.602109 

4.22 

.823990 

7.02 

9 

.537041 

4.30 

.799158 

7.02 

9 

.602362 

4.20 

.824411 

7.03 

10 

.587299 

4.30 

.799579 

7.02 

10 

.602614 

4.20 

.824833 

7.02 

11 

9.587557 

4.30 

9.800000 

7.02 

11 

9.602866 

4.22 

9.825254 

7.02 

12 

.587815 

4.30 

.800421 

7.00 

12 

.603119 

4.20 

.825675 

7.02 

13 

.588073 

4.30 

.800841 

7.02 

13 

.603371 

4.20 

.826096 

7.02 

14 

.588331 

4.28 

.801262 

7.02 

14 

.603623 

4.20 

.826517 

7.02 

15 

.588588 

4.28 

.801683 

7.02 

15 

.60:3875 

^4.20 

.826938 

7.03 

16 

.588846 

4.28 

.802104 

7.00 

!  16 

.604127 

4.20 

.827360 

7.02 

17 

.583103 

4.30 

.802524 

7.02 

17 

.604379 

4.20 

.827781 

7.02 

18 

.580351 

4.28 

.802945 

7.02 

18 

.604631 

4.20 

.828202 

7.02 

19 

.589618 

4.28 

.803366 

7.02 

19 

.604883 

4.18 

.828623 

7.02 

20 

.589875 

4.28 

.803787 

7.00 

20 

.605134 

4.20 

.829044 

7.03 

21 

9.590132 

4.28 

9.804207 

7.02 

21 

9.605386 

4.18 

9.829466 

7.02 

22 

.590389 

4.28 

.804628 

7.02 

22 

.605637 

4.18 

.829887 

7.02 

23 

.593646 

4.28 

.805049 

7.02 

23 

.605888 

4.20 

.830308 

7.02 

24 

.590903 

4.23 

.805470 

7.02   24 

.606140 

4.18 

.830729 

7.03 

25 

.591160 

4.27 

.805891 

7.00 

25 

.606391 

4.18 

.831151 

7.02 

26 

.591416 

4.28 

.806311 

7.02 

26 

.606642 

4.18 

.831572 

7.02 

27 

.591673 

4.27 

.806732 

7.02 

27 

.606893 

4.18 

.831893 

7.03 

28 

.591929 

4.27 

.807153 

7.02 

28 

.607144 

4.17 

.832415 

7.02 

29 

.592185 

4.28 

.807574 

7.02 

29 

.607394 

4.18 

.832836 

7.02 

30 

.592443 

4.27 

.807995 

7.00 

30 

.607645 

4.18 

.833257 

7.03 

31 

9.592698 

4.27 

9.808415 

7.02 

31 

9.607896 

4.17 

9.833679 

7.02 

32 

.592954 

4.27 

.808836 

7.02 

32 

.608146 

4.18 

.834100 

7.03 

33 

.593210 

4.27 

.809257 

7.02 

33 

.608397 

4.17 

.834522 

7.02 

34 

.593466 

4.25 

.809678 

7.02 

34 

.608647 

4.17 

.834943 

7.02 

35 

.593721 

4.27 

.810099 

7.02 

35 

.608897 

4.17 

.835364 

7.03 

36 

.593977 

4.27 

.810520 

7.00 

36 

.609147 

4.17 

.835786 

7.02 

37 

.594233 

4.25 

.810940 

7.02 

37 

.609397 

4.17 

.836207 

7.03 

38 

.594488 

4.25 

.811361 

7.02 

38 

.609647 

4.17 

.836629 

7.02 

39 

.594743 

4.27 

.811782 

7.02 

39 

.609897 

4.17 

.837050 

7.03 

40 

.594999 

4.25 

.812203 

7.02 

40 

.610147 

4.17 

.837472 

7.02 

41 

9.595254 

4.25 

9.812624 

7.02 

41 

9.610397 

4.15 

9.837893 

7.03 

42 

.595509 

4.25 

.813045 

7.02 

42 

.610646 

4.17 

.838315 

7.02 

43 

.595761 

4.25 

.813466 

7.00 

43 

.610896 

4.15 

.838736 

7.03 

44 

.596019 

4.25 

.813886 

7.  l>2 

44 

.611115 

4.15 

.839158 

7.02 

45 

.596274 

4.23 

.814307 

7.02 

45 

.611394 

4.17 

.839579 

7.03 

46 

.596528 

4.25 

.814728 

7.02 

46 

.611644 

4.15 

.840001 

7.03 

47 

.596783 

4.25 

.815149 

7.02 

47 

.611893 

4.15 

.840423 

7.02 

48 

.597038 

4.23 

.815570 

7.02 

48 

.612142 

4.15 

.840844 

7.03 

49 

.597292 

4.23 

.815991 

7.02 

49 

.612391 

4.15 

.841266 

7.02 

50 

.597546 

4.25 

.816412 

7.02 

50 

.612640 

4.13 

.841687 

7.03 

51 

9.597801 

4.23 

9.816833 

7.02 

51 

9.612S88 

4.15 

9.842109 

7.03 

52 

.598055 

4.23 

.817254 

7.02   52 

.613137 

4.15 

.842531 

7.03 

53 

.598309 

4.23 

.817675 

7.02 

53 

.613386 

4.13 

.842953 

7.02 

54 

.598563 

4.23 

.818096 

7.02 

54 

.613634 

4.15 

.843374 

7.03 

55 

.598817 

4.23 

.818517 

7.02 

55 

.613883 

4.13 

.843796 

7.03 

56 

.599071 

4.22 

.818938 

7.02 

56 

.614131 

4.13 

.844218 

7.02 

57 

.599324 

4.23 

.819359 

7.02 

57 

.614379 

4.13 

.844639 

7.03 

58 

.599578 

4.22 

.819780 

7.02 

58 

.614627 

4.15 

.845061 

7.03 

59 

.599831 

4.23 

.820201 

7.02 

59 

.614876 

4.13 

.845483 

7.03 

60  9.600085 

4.22 

9.820622 

7.02 

60 

9.615124 

4.12  9.845905 

7.03 

430 


AND  EXTERNAL  SECANTS. 


54° 

55° 

' 

Vers. 

D.  1". 

Ex.  sec. 

D.  1". 

Vers. 

D.  1'. 

Ex.  sec. 

D.  1'. 

0 

9.615124 

4.12 

9.845905 

7.03 

0 

9.629841 

.05 

9.871250 

7.05 

1 

.615371 

4.13 

.846327 

7.03 

1 

.630084 

.03 

.871673 

7.05 

2 

.615619 

4.13 

.846749 

7.02 

2 

.630326 

.05 

.872096 

7.05 

3 

.615867 

4.13 

.847170 

7.03 

3 

.630569 

.03 

.872519 

.05 

4 

.616115 

4.12 

.847592 

7.03 

4 

.6:30811 

.05 

.872943 

.07 

5 

.616362 

4.13 

.848014 

7.03 

5 

.631054 

4.03 

.873366 

.05 

G 

.616610 

4.12 

.848436 

7.03 

6 

.631296 

4.03 

.873789 

.05 

7 

.616857 

4.12 

.848858 

7.03 

7 

.631538 

4.03 

.874212 

.07 

0 

.617104 

4.12 

.849280 

7.03 

8 

.631780 

4.03 

.874636 

.05 

0 

.017351 

4.13 

.849702 

7.03 

9 

.632022 

4.03 

.875059 

.05 

10 

.617599 

4.10 

.850124 

7.03 

10 

.632264 

4.03 

.875482 

.07 

11 

9.617845 

4.12 

9.850546 

7.03 

11 

9.632505 

4.03 

9.875906 

.05 

13 

.618092 

4.12 

.850968 

7.03 

12 

.632747 

4.03 

.876329 

.05 

13 

.618339 

4.12 

.a51390 

7.03 

13 

.632989 

4.02 

.876752 

.07 

14 

.618586 

4.12 

.851812 

7.03 

14 

.633230 

4.03 

.877176 

.05 

15 

.618833 

4.10 

.852234 

7.03 

15 

.633472 

4.02 

.877599 

.07 

16 

.619079 

4.12 

.852656 

7.03 

16 

.633713 

4.02 

.878023 

.05 

17 

.619326 

4.10 

.853078 

7.03 

17 

.633954 

4.03 

.87B446 

.07 

18 

.019572 

4.10 

.853500 

7.05 

18 

.634196 

4.02 

.878870 

.07 

19 

.619818 

4.12 

.853923 

7.03 

19 

•  .634437 

4.02 

.879294 

.05 

20 

.620065 

4.10 

.854345 

7.03 

20 

.634678 

4.03 

.879717 

.07 

21 

9.620311 

4.10 

9.854767 

7.03 

21 

9.634919 

4.00 

9.880141 

.07 

22 

.620557 

4.10 

.855189 

7.05 

22 

.635159 

.02 

.880565 

.05 

23 

.620803 

4.08 

.855612 

7.03 

!  23 

.635400 

.02 

.880988 

.07 

24 

.621048 

4  10 

.856034 

7.03 

!  24 

.635641 

.00 

.881412 

7.07 

25 

.621294 

4.10 

.856456 

7.03 

25 

.635881 

.02 

.881836 

7.07 

26 

.621540 

4.10 

.856878 

7.05 

26 

.636122 

.00 

.882260 

7.05 

27 

.621786 

4.08 

.857301 

7.03 

27 

.636362 

.02 

.882683 

7.07 

23 

.622031 

4.08 

.857723 

7.03 

28 

.636603 

.00 

.883107 

7.07 

29 

.622276 

4.10 

.858145 

7.05 

29 

.636843 

.00 

.883531 

7.07 

30 

.622522 

4.08 

.858568 

7.03 

30 

.637083 

.00 

.883955 

7.07 

31 

9.622767 

4.08 

9.858990 

7.05 

31 

9.637323 

4.00 

9.884379 

7.07 

32 

.623012 

4.08 

.859413 

7.03 

32 

.637563 

4.00 

.884803 

7.07 

33 

.623257 

4.08 

.859835 

7.05 

33 

.637803 

4.00 

.885227 

7.07 

34 

.623502 

4.08 

.860258 

7.03 

34 

.638043 

4.00 

.885651 

7.07 

35 

.623747 

4.08 

.86068-3 

7.05 

35 

.638283 

3.98 

.886075 

7.07 

36 

.623992 

4.08 

.861103 

7.03 

36 

.638522 

4.00 

.886499 

7.07 

37 

.624237 

4.07 

.861525 

7.05 

37 

.638762 

3.98 

.886923 

7.07 

38 

.624481 

4.08 

.8(51948 

7.03 

38 

.639001 

4.00 

.887347 

7.08 

39 

.624726 

4.07 

.862370 

7.05 

.639241 

3.98 

.887772 

7.07 

40 

.624970 

4.08 

.862793 

7.03 

40 

.639480 

3.98 

.888196 

7.07 

41 

9.625215 

4  07 

9.863215 

7.05 

41 

9.639719 

3.98 

9.888620 

7.07 

42 

.035459 

4.07 

.863638 

7.05 

42 

.639958 

3.98 

.  .88904-1 

7.08 

43 

.625703 

4.07 

.864061 

7.03 

43 

.640197 

3.98 

.88Q469 

7.07 

41 

.625947 

4.07 

.864483 

7.05 

44 

.640436 

3.98 

.889893 

7.07 

45 

.626191 

4.07 

.864906 

7.05 

45 

.640675 

3.98 

.890317 

7.08 

46 

.626435 

4.07 

.865329 

7.05 

!  46 

.640914 

3.98 

.890742 

7.07 

47 

.626679 

4.07 

.865752 

7.03 

1  47 

.641153 

3.97 

.891166 

7.08 

48 

.626923 

4.05 

.866174 

7.05 

!  48 

.641391 

8.98 

.891591 

7.07 

49 

.627166 

4.07 

.86(5597 

7.05 

|  49 

.641630 

3.97 

.892015 

7.08 

50 

.627410 

4.07 

.867020 

7.05 

50 

.641868 

3.98 

.892440 

7.07 

51 

9.627654 

4.05 

9.867443 

7.05 

51 

9.642107 

3.97 

9.892864 

7.08 

52 

.627897 

4.05 

.867866 

7.05 

52 

.642:545 

3.97 

.893289 

7.08 

53 

.628140 

4.07 

-  .868289 

7.05 

53 

.642583 

3.98 

.893714 

7.07 

54 

.628384 

4.05 

.868712 

7.05 

54 

.642822 

3.97 

.894138 

7.08 

55 

.623627 

4.05 

.869135 

7.05 

55 

.643060 

3.97 

.894563 

7.08 

56 

.628870 

4.05 

.869558 

7.05 

56 

.643298 

3.95 

.894988 

7.07 

57 

.629113 

4.05 

.869981 

7.05 

57 

.643535 

3.97 

.895412 

7.08 

58 

.629356 

4.03 

.870404 

7.05  ! 

58 

.643773 

3.97 

.895837 

7.08 

59 

.629598 

4.05 

.870827 

7.05  | 

59 

.644011 

3.97 

.896262 

7.08 

60 

9.629841 

4.05 

9.871250 

7.05  i 

60 

9.644249 

3.95 

9.896687 

7.08 

431 


TABLE  XXVI. -LOGARITHMIC  VERSED  SINES 


56° 

I 

1 

57° 

/ 

Vers. 

D.  1'. 

Ex.  sec. 

D  1". 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  1". 

0 

9.644249 

3.95 

9.896687 

7.08 

0 

9.658356 

3.87 

9.922247 

7.12 

1 

.644486 

8.97 

.897112 

7.08 

1 

.658588 

3.88 

.922674 

7.13 

2 

.044724 

8.1)5 

.897537 

7.08 

2 

.658821 

3.87 

.923102 

7.12 

3 

.644961 

3.95 

.897962 

7.08 

3 

.659053 

3.88 

.923529 

7.12 

4 

.645198 

8.95 

.898387 

7.08 

4 

.659286 

3.87 

.923956 

7.13 

5 

.645435 

8.97 

.898812 

7.08 

5 

.659518 

3.87 

.924384 

7.12 

6 

.645673 

3.95 

.899237 

7.08 

6 

.659750 

3.88 

.924811 

7.13 

7 

.645910 

3.95 

.899662 

7.08 

7 

.659983 

3.87 

.9252S9 

7.12 

8 

.646147 

3.95 

.900087 

7.08 

8 

.660215 

3.87 

.925GG6 

7.13 

9 

.646384 

3.93 

.900512 

7.10 

9 

.660447 

3.87 

.926094 

7.12 

10 

.646620 

3.95 

.900938 

7.08 

10 

.660679 

3.85 

.926521 

7.13 

11 

9.646857 

3.95 

9.901363 

7.08 

11 

9.660910 

3.87 

9.926949 

7.13 

12 

.647094 

3.93 

.901788 

7.08 

12 

.661142 

3.87 

.927377 

7.12 

13 

.647330 

3.95 

.902213 

7.10 

13 

.661374 

3.85 

.927804 

7.13 

14 

.647567 

3.93 

.902039 

7.08 

14 

.661605 

3.87 

.928232 

7.13 

15 

.647803 

3.93 

.908064 

7.10 

15 

.661837 

-3.85 

.928060 

7.13 

16 

.648039 

3.95 

.903490 

7.08 

16 

.662068 

3.87 

.929088 

7.13 

17 

.648276 

3.93 

.903915 

7.10 

17 

.662300 

8.85 

.929516 

7.13 

18 

.648512 

3.93 

.904341 

7.08 

18 

.662531 

3.85 

.929944 

7.13 

19 

.648748 

3.93 

.  .904766 

7.10 

19 

.662762 

3.85 

.930372 

7.13 

20 

.648984 

3.93 

.905192 

7.08 

20 

.662993 

3.85 

.930800 

7.13 

21 

9.649220 

3.93 

9.905617 

7.10 

21 

9.663224 

3.85 

9.931228 

7.13 

22 

.649450 

3.92 

.906043 

7.10 

22 

.003455 

3.85 

.931656 

7.15 

23 

.649691 

3.93 

.906469 

7.08 

23 

.G03G86 

3.85 

.932085 

7.13 

24 

.649927 

3.93 

.906894 

7.10 

24 

.663917 

3.85 

.932513 

7.13 

25 

.650163 

3.92 

.907320 

7.10 

25 

.664148 

3.83 

.932941 

7.13 

26 

.650398 

3.92 

.907746 

7.10 

26 

.664378 

3.85 

.933369 

7.15 

27 

.650633 

3.93 

.908172 

7.10 

27 

.664009 

3.83 

.933798 

7.13 

28 

.650869 

3.92 

.908598 

7.10 

28 

.664839 

3.85 

.934226 

7.15 

29 

.651104 

3.92 

.909024 

7.10 

29 

.665070 

3.  as 

.934655 

7.13 

30 

.651339 

3.92 

.909450 

7.10 

30 

.665300 

3.83 

.935083 

7.15 

31 

9.651574 

3.92 

9.909876 

7.10 

31 

9.665530 

3.83 

9.935512 

7.15 

32 

.651809 

3.92 

.910302 

7.10 

32 

.665700 

3.83 

.935941 

7.13 

33 

.652044 

3.92 

.910728 

7.10 

33 

.665990 

3.83 

.936369 

7.15 

34 

.652279 

3.92 

.911154 

7.10 

34 

.G66220 

3.83 

.936798 

7.15 

35 

.652514 

3.90 

.911580 

7.10 

35 

.666450 

3.83 

.937227 

7.15 

36 

.652748 

3.92 

.912006 

7.10 

36 

.666680 

3.83 

.937656 

7.15 

37 

.052983 

3.90 

.912432 

7  12 

37 

.666910 

3.82 

.938085 

7.13 

38 

.653217 

3.92 

.912859 

7!lO 

38 

.667139 

3.83 

.938513 

7M 

39 

.653452 

3.90 

.913285 

7.10 

39 

.667369 

3.83 

.938942 

7.15 

40 

.653686 

3.90 

.913711 

7.12 

40 

.667599 

3.82 

.939371 

7.17 

41 

9.653920 

3.92 

9.914138 

7.10 

41 

9.667828 

3.82 

9.939801 

7.15 

42 

.654155 

3.90 

.914564 

7.12 

42 

.668057 

3.83 

.940230 

7.15 

43 

.654389 

3.90 

.914991 

7.10 

43 

.668287 

3.82 

.940659 

7.15 

44 

.654623 

3.90 

.915417 

7.12 

44 

.668516 

3.82 

.941088 

7.15 

45 

.654857 

3.88 

.915844 

7.10 

45 

.6687'45 

3.82 

.941517 

7.17 

46 

.655090 

3.90 

.916270 

7.12 

46 

.668974 

3.82 

.941947 

7.15 

47 

.655324 

3.90 

.91GG97 

7.12 

47 

.669203 

3.82 

.94:2370 

7.15 

48 

.655558 

3.90 

.917124 

7.10 

48 

.6G9432 

3.62 

.942806. 

7.15 

49 

.655792 

3.88 

.917550 

7.12 

49 

.G6CGG1 

3.80 

.943235 

7.17 

60 

.056025 

3.88 

.917977 

7.12 

50 

.669889 

3.82 

.943665 

7.15 

51 

9.656258 

8.90 

9.918404 

7.12 

51 

9.670118 

3.82 

9.944094 

7.17 

52 

.65G492 

3.88 

.918831 

7.12 

52 

.670347 

3.80 

.944524 

7.15 

53 

.650725 

8.88 

.910558 

7.12 

53 

.G7'0575 

3.82 

.944953 

7.  7 

54 

.G5GS58 

8.88 

.919385 

7.12 

54 

.670804 

3.80 

.945383 

7.  7 

55 

.657191 

3.88 

.920112 

7.12 

55 

.671032 

3.80 

.945813 

7.  7 

56 

•G?I4?i 

3.88 

.920539 

7.12 

56 

.671260 

3.80 

.946243 

7.  7 

57 

.657G57 

8.88 

.920966 

7.12 

57 

.671488 

3.80 

.94GG73 

7.  7 

58 

.657890 

3.88 

.921393 

7.12 

58 

.671716 

3.82 

.947103 

7.  7 

59 

.658123 

3.88 

.921820 

7.12 

59 

.671945 

3.78 

.947533 

7.  7 

60 

9.658356 

3.87 

9.922247 

7.12 

GO 

9.672172 

3.80 

9.947963 

7.  7 

AND  EXTERNAL  SECANTS. 


58° 

59° 

' 

Vers. 

D.I'. 

Ex.  sec. 

D.I". 

/ 

Vers. 

D.I". 

Ex.  sec. 

D.  r. 

0 

9.G72172 

3.80 

9.947963 

7.17 

0 

9.685703 

3.72 

9.973868 

7.23 

1 

.672400 

3.80 

.948:393 

7.17 

1 

.685931 

3.72 

.974302 

7.23 

2 

.07:3023 

3.80 

.948883 

7.17 

2 

.080154 

3.72 

.974736 

7.22 

3 

.072356 

3.78 

.949253 

7.17 

3 

.686377 

3.72 

.975169 

7.23 

4 

.673083 

3.80 

.949683 

7.18 

4 

.686600 

3.72 

.975003 

7.23 

5 

.673311 

3.78 

.950114 

7.17 

5 

.686823 

3.72 

.976037 

7.23 

6 

.673533 

3.80 

.950544 

7.18 

6 

.687046 

3.72 

.970471 

7.23 

7 

.073766 

3.78 

.950975 

7.17 

7 

.687269 

3.72 

.976905 

7.23 

8 

.673993 

3.78 

.951405 

7.18 

8 

.687492 

3.70 

.977339 

7.23 

9 

.674220 

3.80 

.951836 

7.17 

9 

.687714 

3.72 

.977773 

7.23 

10 

.674448 

3.78 

.952266 

7.18 

10 

.687937 

3.70 

.978207 

7.23 

11 

9.674675 

3.78 

9.952697 

7.18 

11 

9.688159 

3.72 

9.978641 

7.33 

12 

.074902 

3.78 

.953128 

7.17 

12 

.688382 

3.70 

.979075 

7.25 

13 

.075123 

3.78 

.953558 

7.18 

13 

.688604 

3.70 

.979510 

7.23 

14 

.075356 

3.77 

.953989 

7.18 

14 

.688826 

3.70 

.979944 

7.25 

15 

.075582 

3.78 

.954420 

7.18 

15 

.689048 

3.72 

.980379 

7.23 

16 

.075809 

3.78 

.954851 

7.18 

16 

.089271 

3.70 

.980813 

7.25 

17 

.076030 

3.77 

.955282 

7.18 

17 

.689493 

3.70 

.981248 

7.23 

18 

.676262 

3.78 

.95.-)?  13 

7.18 

18 

.689715 

3.70 

.981682 

7.25 

19 

.670489 

3.77 

.956144 

7.18 

19 

.689937 

3.08 

.982117 

7.25 

20 

.076715 

3.77 

.956575 

7.18 

20 

.690158 

3.70 

.982552 

7  25 

21 

9.676941 

3.78 

9.957006 

7.20 

21 

9.G90380 

3.70 

9.982987 

7.25 

2:2 

.077108 

3.77 

.95i'438 

7.18 

22 

.690602 

3.08 

.983422 

7.25 

23 

.077394 

3.77 

.957869 

7.18 

23 

.690823 

3.70 

.983857 

7.25 

24 

.  077020 

3.77 

.958300 

7.20 

;  24 

.691045 

3.08 

.984292 

7.25 

23 

.677846 

3.77 

.953732 

7.18 

|  25 

.691266 

3.70 

.984727 

7.25 

20 

.678072 

3.77 

.959163 

7.20 

1  26 

.691488 

3.08 

.985162 

7.25 

27 

.678298 

3.75 

.939595 

7.18 

27 

.691709 

3.08 

.985597 

7.27 

23 

.078523 

3.77 

.960026 

7.20 

28 

.691930 

3.08 

.986033 

7.25 

29 

.078749 

3.77 

.980458 

7.20 

29 

.692151 

3.08 

.980408 

7.27 

30 

.078975 

3.75 

.9(30390 

7.18 

!  30 

.692372 

3.68 

.986904 

7.25 

31 

9.679200 

3.77 

9.961321 

7.20 

31 

9.692593 

3.68 

9.987339 

7.27 

32 

.079426 

3.75 

.901753 

7.20 

32 

.692814 

3.68 

.987775 

7.25 

33 

.679651 

3.75 

.962185 

7.20 

!  33 

.693035 

3.08 

.988210 

7.27 

34 

.079  -(76 

3.77 

.962617 

7.20 

34 

.693256 

3.08 

.988646 

7.27 

35 

.680103 

3.75 

.963049 

7.20 

35 

.693477 

3.67 

.989082 

7.27 

36 

.080327 

3.75 

.963481 

7.20 

36 

.693697 

3.68 

.989518 

7.27 

37 

.680552 

3.75 

.963913 

7.20 

37 

.693918 

3.67 

.989954 

7.27 

38 

.680777 

3.75 

.964345 

7.22 

38 

.694138 

3.08 

.990390 

7.27 

39 

.681002 

3.75 

.964778 

7.20 

39 

.694359 

3.07 

.990826 

7.27 

40 

.681227 

3.73 

.955210 

7.20 

40 

.694579 

3.67 

.991262 

7.27 

41 

9.681451 

3.75 

9.905642 

7.22 

41 

9.694799 

3.67 

9.991698 

7.27 

42 

.681<i7'6 

3.75 

.966075 

7.20 

i  42 

.695019 

3.68 

.992134 

7.28 

43 

.681901 

3.73 

.966507 

7.22 

43 

.695240 

3.67 

*  .992571 

7.27 

44 

.682125 

3.75 

.960940 

7.20 

44 

.695460 

3.67 

.99:3007 

7.28 

45 

.682350 

3.73 

.967372 

7  22 

45 

.695080 

3.65 

.993444 

7.27 

46 

.682574 

3.73 

.967805 

riaa 

46 

.095899 

3.67 

.993880 

7.28 

47 

.682798 

3.75 

.968238 

7.20 

47 

.690119 

3.67 

.994317 

7.28 

48 

.683023 

3.73 

.908670 

7.22 

48 

.096339 

3.67 

.994754 

7.28 

49 

.683247 

3.73 

.909103 

7.22 

!  49 

.696559 

3.65 

.995191 

7.27 

50 

.683471 

3.73 

.909536 

7.22 

50 

.69677-8 

3.67 

.995627 

7.28 

51 

9.683695 

3.73 

9.969969 

7.22 

51 

9.096998 

3.65 

9.996064 

7.28 

52 

.683919 

3.73 

.970402 

7.22 

!  52 

.697217 

3.07 

.996501 

7.28 

r.a 

.684143 

3.73 

.970835 

7.22 

!  53 

.697437 

3.05 

.990938 

7.  SO 

54 

.684307 

3.72 

.971268 

7.22 

!  54 

.697656 

3.  05 

.997376 

7.28 

55 

.684590 

3.73 

.971701 

7.23  ]  55 

.697875 

3.05 

.997813 

7.28 

56 

.684814 

3.72 

.972135 

7.22   56 

.098094 

3.05 

.998250 

7.28 

5? 

.685037 

3.73 

.972568 

7.22   57 

.698313 

3.05 

.998687 

7.30 

58 

.685261 

3.72 

.973001 

7.23  1  58 

.698532 

3.65 

.999125 

7.28 

f5U 

.685484 

3.73 

.973435 

7.22  !!  59 

.698751 

3.65 

9.999562 

7.30 

CO 

9.  685708 

3.72 

9.973868 

7.23  ll  60 

9.698970 

3.63 

10.000000 

7.30 

433 


TABLE  XXVI.-LOGARITHMIC  VERSED  SINES 


60° 

61° 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.I', 

/ 

Vers. 

D.r. 

Ex.  sec. 

D.l". 

0 

9.698970 

3.65 

10.000000 

7.30 

0 

9.711968 

3.57 

10.026397 

7.37 

1 

.699189 

3.63 

.000438 

7.28 

1 

.712182 

3.58 

.026&39 

7.37 

2 

.699407 

3.65 

.000875 

7.30 

2 

.712397 

3.57 

.027281 

7.38 

3 

.699626 

3.65 

.001313 

7.30 

3 

.712611 

3.57 

.027724 

7.38 

4 

.699845 

3.63 

.001751 

7.30 

4 

.712825 

3.57 

.028167 

7.37 

5 

.700063 

3.65 

.002189 

7.30 

5 

.713039 

3.57 

.028609 

.38 

6 

.700282 

3.63 

.002627 

7.30 

6 

.713253 

3.57 

.029052 

.38 

7 

.700500 

3.63 

.003065 

7.30 

7 

.713467 

3.57 

.029495 

.38 

8 

.700718 

3.63 

.003303 

7.32 

8 

.713681 

3.57 

.029938 

.38 

9 

.700936 

3.63 

.003942 

7.30 

9 

.713895 

3.57 

.030381 

.40 

10 

.701154 

3.63 

.004380 

7.30 

10 

.714109 

3.57 

.030825 

.38 

11 

9.701372 

3.63 

10.004818 

7.32 

11 

9.714323 

3.55 

10.031268 

"•.38 

12 

.701590 

3.63 

.005257 

7.30 

12 

.714536 

3.57 

.031711 

^.40 

13 

.701808 

3.63 

.005695 

7.32 

13 

.714750 

3.55 

.032155 

".38 

14 

.702026 

3.63 

.006134 

7.32 

14 

.714963 

3.57 

.032598 

".40 

15 

.702244 

3.63 

.006573 

7.32 

15 

.715177 

~  3.55 

.033042 

".40 

10 

.702462 

3.62 

.007012 

7.30 

16 

.715390 

3.55 

.033486 

7.38 

17 

.702679 

3.63 

.007450 

7.32 

17 

.715603 

3.57 

.033929 

7.40 

18 

.702897 

3.62 

.007889 

7.32 

18 

.715817 

3.55 

.034373 

.40 

19 

.703114 

3.63 

.008328 

7.32 

19 

.716030 

3.55 

.034817 

.40 

20 

.703332 

3.62 

.008767 

7.33  j 

20 

.716243 

3.55 

.035261 

.40 

21 

9.703549 

3.62 

10.009207 

7.32 

21 

9.716456 

3.55 

10.035705 

.42 

22 

.703766 

3.62 

.009646 

7.32 

22 

.716669 

3.55 

.036150 

.40 

23 

.703983 

3.62 

.010085 

7.33 

23 

.716882 

8.55 

.036594 

.40 

24 

.704200 

3.62 

.010525 

7.32 

24 

.717095 

3.53 

.037038 

7.42 

25 

.704417 

3.62 

.010964 

7.33 

25 

.717307 

3.55 

.037483 

7.42 

26 

.704634 

3.62 

.011404 

7.32 

26 

.717520 

3.53 

.037928 

7.40 

27 

.704851 

3.62 

.011843 

7.33 

27 

.717732 

3.55 

.038372 

7.42 

28 

.705068 

3.62 

.012283 

7.33 

28 

.717945 

3.53 

.038817 

7.42 

29 

.705285 

3.60 

.012723 

7.33 

29 

.718157 

3.55 

.039262 

7  !43 

30 

.705501 

3.62 

.013163 

7.33 

30 

.718370 

3.53 

.039707 

7.42 

31 

9.705718 

3.62 

10.013603 

7.  as 

31 

9.718582 

3.53 

10.040152 

7.42 

32 

.705935 

3.60 

.014043 

7  33 

32 

.718794 

3.55 

.040597 

7.42 

33 

.706151 

3.60 

.014483 

7^33 

33 

.719007 

3.53 

.041042 

7.43 

34 

.706367 

3.62 

.014923 

7.33 

34 

.719219 

3.53 

.041488 

7.42 

35 

.706584 

3.60 

.015363 

7.35 

35 

.719431 

3.53 

.041933 

7.43 

36 

.706800 

3.60 

.015804 

7.33 

36 

.719643 

3.53 

.042379 

7.42 

37 

.707016 

3.60 

.016244 

7.33 

37 

.719855 

3.52 

.042824 

7!  43 

38 

.707232 

3.60 

.016684 

7.33 

38 

.720066 

3.53 

.043270 

7.1:) 

39 

.707448 

3.60 

.017125 

7.35 

39 

.720278 

3.53 

.043716 

7.43 

40 

.707664 

3.60 

.017566 

7.35 

40 

.720490 

3.52 

.044162 

7.43 

41 

9.707880 

3.60 

10.018007 

7.33 

41 

9.720701 

3.53 

10.044608 

7.43 

43 

.708096 

3.58 

.018447 

7.35 

42 

.720913 

3.52 

.045054 

7.43 

43 

.708311 

3.60 

.018888 

7.35 

43 

.721124 

3.53 

.045500 

7.43 

41 

.708527 

3.60 

.019329 

7.35 

44 

.721336 

3.52 

.045946 

7.45 

45 

.708743 

3.58 

.019770 

7.37 

45 

.721547 

3.52 

.046393 

7.43 

46 

.708958 

3.60 

.020212 

7.35 

46 

.721758 

3.53 

•046839 

7.45 

47 

.709174 

3.58 

.020653 

7.35 

47 

.721970 

8.52 

.047286 

7.43 

48 

.709389 

3.58 

.021094 

7.35 

48 

.722181 

3.52 

.047732 

7.45 

49 

.709604 

3.58 

.021535 

7.37 

49 

.722392 

3.52 

.048179 

7.45 

50 

.709819 

3.58 

.021977 

7.37 

50 

.722603 

3.52 

.048620 

7.45 

51 

9.710035 

3.58 

10.022419 

7.35 

51 

9.722814 

3.50 

10.049073 

7.45 

52 

.710250 

3.58 

.022800 

7.37 

52 

.723024 

3.52 

.049520 

7.45 

53 

.710465 

3.58 

.023302 

7.37 

53 

.723235 

3.52 

.049967 

7.45 

54 

.710680 

3.58 

.023744 

7.37 

54 

.723446 

3.52 

.050414 

7.45 

55 

.710895 

3.57 

.024186 

7.37 

55 

.723657 

3.50 

.050861 

.47 

56 

.711109 

3.58 

.024628 

7.37 

56 

.723867 

3.52 

.051309 

7.45 

57 

.711324 

3.58 

.025070 

7.37 

57 

.724078 

3.50 

.051756 

.47 

58 

.711539 

3.57 

.025512 

7.37 

58 

.724288 

8.50 

.052204 

7.47 

59 

.711753 

3.58 

.025954 

7.38 

59 

.724498 

3.52 

.052652 

.45 

60 

9.711968 

3.57 

10.026397 

7.37 

60 

9.724709 

3.50 

10.053099 

7.47 

.  434 


AND  EXTERNAL  SECANTS. 


62° 

63° 

/ 

Vers. 

D.  1. 

Ex.  sec. 

D.  1'. 

/ 

Vers. 

D.I'. 

Ex.  sec. 

D.r. 

0 

9.724709 

3.50 

10.053099 

7.47 

0 

9.737200 

3.43 

10.080153 

7.58 

1 

.724919 

3.50 

.058547 

7.47 

i 

.737406 

3.43 

.080608 

7.57 

2 

.725129 

3.50 

.053995 

7.47 

2 

.737612 

3.43 

.081062 

7.57 

3 

.725339 

3.50 

.054443 

7.48 

8 

.737818 

3.43 

.081516 

7.58 

4 

.725549 

3.50 

.054892 

7.47 

4 

.738024 

3.43 

.081971 

7.57 

5 

.725759 

3.50 

.055340 

7.47 

5 

.738230 

3.43 

.082425 

7.58 

6 

.725969 

3.50 

.055788 

7.48 

6 

.738436 

3.43 

.082880 

7.58  | 

7 

.726179 

3.48 

.056237 

7.47  i 

7 

.738642 

3.42 

.083335 

7.58 

8 

.726388 

3.50 

.056685 

7.48  1  8 

.738847 

3.43 

.083790 

7.58 

9 

.726598 

3.50 

.057134 

7.48  1  9 

.739053 

3.42 

.084245 

7.58 

10 

.726808 

3.48 

.057583 

7.48 

10 

.739258 

3.43 

.084700 

7.58 

11 

9.727017 

3.50 

10.058032 

7.48  I 

11 

9.739464 

3.42 

10.085155 

7.60 

12 

.727227 

3.48 

.0.58481 

7.48 

12 

.739669 

3.43 

.085611 

7.58 

13 

.727436 

3.48 

.058930 

7.48 

13 

.739875 

3.42 

.086C66 

7.60 

14 

.727645 

3.50 

.059379 

7.48  !  14 

.740080 

3.42 

.086522  7.58 

15 

.727855 

3.48 

.059828 

7.50  i  15 

.740285 

3.42 

.086977  7.60 

16 

.728064 

3.48 

.060278 

7.48  !  16 

.740490 

3.42 

.087433  7.60 

17 

.728273 

3.48 

.060727 

7.50  ! 

17 

.740695 

3.42 

.087889  i  7.60 

18 

.728482 

3.48 

.061177 

7.48  ! 

18 

.740900 

3.42 

.088345  7.60 

19 

.728691 

3.48 

.061626  7.50  1 

19 

.741105 

3.42 

.088801   7.62 

20 

.728900 

3.48 

.062076   7.50  ! 

20 

.741310 

3.42 

.089258 

7.60 

21 

9.729109 

3.47 

10.062526  7.50 

21 

9.741515 

3.40 

10.089714 

7.62 

22 

.729317 

3.48 

.062976 

7.50  !  23 

.741719 

3.42 

.090171 

7.60 

23 

.729526 

3.43 

.063425 

7.50  1 

23 

.741924 

3.42 

.090627 

7.62 

21 

.729735 

3.4V 

.063876 

7.52 

24 

.742129 

3.40 

.091084 

7.62 

25 

.729943 

3.48 

.064327 

7.50 

25 

.7'42333 

3.42 

.091541 

7.62 

26 

.730152 

8.«¥ 

.064777 

7.50  ! 

26 

.7-42538 

3.40 

.091998 

7.62 

27 

.730360 

3.48 

.065227 

7.52  j 

27 

.742742 

3.40 

.092455 

7.62 

28 

.730569 

3.47 

.065678 

7.52 

28 

.742946 

3.40 

.092912 

7.63 

29 

.730777 

3.47 

.066129 

7.52  ! 

29 

.743150 

3.42 

.093370 

7.62 

30 

.730985 

3.47 

.066580 

7.50 

30 

.743355 

3.40 

.093827 

7.63 

31 

9.731193 

3.47 

10.067030 

7.53 

81 

9.7'43559 

3.40 

10.094285 

7.63 

32 

.731401 

3.47 

.067482 

7.52 

32 

.743763 

3.40 

.094743 

7.62 

33 

.731609 

3.47 

.067933 

7.52 

83 

.743967 

3.40 

.095200 

7.63 

34 

.731817 

3.47 

.06&384 

7.52 

34 

.744171 

3.40 

.095658 

7.63 

35 

.732025 

3.47 

.068835 

7.53 

35 

.744375 

3.38 

.096116 

7.65 

36 

.733233 

3.47 

.069287 

7.52 

36 

.744578 

3.40 

.096575 

7.63 

37 

.732441 

3.45 

.0697:38 

7.53 

37 

.744782 

3.40 

.097'033 

7.63 

38 

.732648 

3.47 

.070190 

7.53 

38 

.744986 

3.38 

.097491 

7.65 

39 

.732856 

3.47 

.070642 

7.52 

39 

.745189 

3.40 

.097950 

7.63 

43 

.733064 

3.45 

.071093 

7.53 

40 

.7'45393 

3.38 

.098408 

7.65 

41 

9.7&3271 

3.45 

10.071545 

7.55 

41 

9.7'45596 

3.40 

10.098867 

7.65 

42 

.733478 

3.47 

.071998 

7.53 

42 

.745800 

3.38 

.099326 

7.65 

43 

.733686 

3.45 

.072450 

7.53 

43 

.746003 

3.38 

-  .099785 

7.65 

44 

.733893 

3.45 

.072902 

7.53 

44 

.746206 

3.38 

.100244 

7.67 

45 

.734100 

3.45 

.073354 

7.55 

45 

.746409 

3.40 

.100704 

7.65 

46 

.734307 

3.47 

.073807 

7.55 

46 

.746613 

3.38 

.101163 

7.67 

47 

.734515 

3.43 

.074260 

7.53 

47 

.746816 

3.38 

.101623 

7.65 

48 

.734721 

3.45 

.074712 

7.55 

48 

.747-019 

3.38 

.102082 

7.67 

48 

.734028 

3.45 

.075165 

7.55 

49 

.747222 

3.37 

.102542 

7.67 

50 

.735135 

3.45 

.075618 

7.55 

50 

.747424 

3.38 

.103002 

7.67 

51 

9.735342 

3.45 

10.076071 

7.55 

51 

9.747627 

3.38 

10.103462 

7.67 

52 

.735549 

3.43 

.076524 

7.55 

52 

.7478:30 

3.38 

.103922 

7.67 

53 

.735755 

3.45 

.076977 

7.57 

53 

.748033 

3.37 

.104382 

7.68 

54 

.735962 

3.45 

.077431 

7.55 

54 

.748235 

3.38 

.104843 

7.67 

55 

.736169 

3.43 

.077884 

7.57 

55 

.748438 

3.37 

.105303 

7.68 

56 

.736375 

3.43 

!  078838 

7.57 

56 

.748640 

3.38 

.105764 

7.67 

57 

.730581 

3.45 

.078792 

7.55 

57 

.7'48843 

3.37 

.106224 

7.68 

58 

.736788 

3.43 

.079245 

7.57 

58   .749045 

3.37 

.100685 

7.68 

59 

.736994 

3.43 

.079699 

7.57 

59 

.749247 

3.37 

.107146 

7.68 

60 

9.737200 

3.43 

10.080153 

7.58 

60 

9.749449 

3.38 

10.107607 

7.70 

435 


TABLE  XXVI. -LOGARITHMIC  VERSED  SINES 


64 

0 

I 

65° 

f 

Vers. 

D.  1". 

Ex.  sec. 

D  r. 

/ 

Vers. 

D.  r. 

Ex.  sec. 

D.I". 

o 

9.749449 

3.38 

10.107607 

7.70 

0 

9.761463 

3.30 

10.135515 

7.82 

1 

.749652 

3.37 

.108069 

7.68 

1 

.761661 

3.32 

.135984 

7.83 

2 

.749854 

3.37 

.108530 

7.70 

2 

.761800 

3.30 

.136454 

.82 

3 

.750056 

3.37 

.108992 

7.68 

3 

.762058 

3.30 

.136923 

.83 

4 

.750258 

3.35 

.109453 

7.70 

4 

.762256 

3.30 

.137393 

.83 

5 

.750459 

3.37 

.109915 

7.70 

5 

.762454 

3.30 

.137863 

.83 

6 

.750661 

3.37 

.110377 

7.70 

6 

.762652 

3.30 

.138333 

.83 

7 

.750863 

3.37 

.110839 

7.70 

7 

.762850 

3.28 

.138803 

.83 

8 

.751065 

3.35 

.111301 

7.70 

8 

.763047 

3.30 

.139273 

.85 

9 

.751266 

3.37 

.111763 

7.72 

9 

.763245 

3.30 

.139744 

.83 

10 

.751468 

3.35 

.112226 

7.70 

10 

.763443 

3.30 

.140214 

.85 

11 

9.751669 

3.37 

10.112688 

7.72 

11 

9.763641 

3.28 

10.140685 

.85 

12 

.751871 

3.35 

.113151 

7.72 

12 

.763838 

3.30 

.141156 

.85 

13 

.752072 

3.35 

.113614 

7.72 

13 

.764036 

3.28 

.141627 

.85 

14 

.752273 

3.37 

.114077 

7.72 

14 

.764233 

3.28 

.142098 

.85 

15 

.752475 

3.35 

.114549 

7.72 

15 

.764430 

"3.30 

.142569 

.87 

1G 

.752676 

3.35 

.115003 

16 

.764628 

3.28 

.143041 

.85 

17 

.752877 

3.35 

.115466 

7:72 

17 

.764825 

3.28 

.143512 

.87 

18 

.75:3078 

3.35 

.115929 

7.73 

18 

.765022 

3.28 

.143984 

.87 

19 

.753279 

3.35 

.116393 

7.73 

19 

.765219 

3.28 

.144456 

.87 

20 

.753480 

3.35 

.116857 

7.73 

20 

.765416 

3.28 

.144928 

.87 

21 

9.753681 

3.33 

10.117321 

7.73 

21 

9.765613 

3.28 

10.145400 

.87 

22 

.753881 

3.35 

.1177&5 

7.73 

22 

.765810 

3.28 

.145873 

.88 

23 

.754082 

3.35 

.118249 

7.73 

23 

.766007 

3.28 

.146345 

.88 

24 

.754283 

3.33 

.118713 

7.73 

24 

.766204 

3.28 

.146818 

.87 

25 

.754483 

3.35 

.119177 

7.75 

25 

.766401 

3.27 

.147290 

.88 

26 

.754684 

3.33 

.119642 

7.73 

26 

.766597 

3.28 

.147763 

.88 

27 

.754884 

3.35 

.120106 

7.75 

27 

.766794 

3.28 

.148236 

.90 

28 

.755085 

3.33 

.120571 

7.75 

28 

.766991 

3.27 

.148710 

.88 

29 

.755285 

3.33 

.121036 

7.75 

29 

.767187 

3.28 

.149183 

.00 

30 

.755485 

3.33 

.121501 

7.75 

30 

.767384 

3.27 

.149657 

.88 

81 

9.755685 

3.35 

10.121966 

7.75 

31 

0.767580 

3.27 

10.150130 

.DO 

32 

.755886 

3  33 

.1^2431 

7.77 

32 

.767776 

3.27 

.150604 

.00 

33 

.756086 

300 
.00 

.122897 

7.75 

33 

.767972 

3.28 

.151078 

.SO 

34 

.756286 

3.  as 

.123362 

7.77 

34 

.768169 

3.27 

.151552 

.C2 

35 

.756486 

3.32 

.12:3828 

7.77 

35 

.768365 

3.27 

.152027 

.90 

36 

.756685 

3.33 

.124294 

7.77 

36 

.768561 

3.27 

.152501 

.92 

37 

.756885 

3.  ,33 

.124760 

7.77 

37 

.768757 

3.27 

.152976 

.90 

38 

.757085 

3.33 

.125226 

7.77 

38 

.768953 

3.27 

.153450 

.92 

39 

.757285 

3.32 

.125692 

7.77 

39 

.769149 

3.25 

.153925 

.92 

40 

.757484 

3.33 

.126158 

7.78 

40 

.769344 

3.27 

.154400 

.93 

41 

9.757684 

3.32 

10.126625 

7.78 

41 

9.769540 

3.27 

10.154876 

.98 

42 

.757883 

3.33 

.127092 

7.77 

42 

.769736 

3.25 

.155351 

.1)2 

43 

.758083 

3.32 

.  127558 

7.78 

43 

.769931 

3.27 

.155826 

.93 

44 

.758282 

3.32 

.128025 

7.78 

44 

.770127 

3.27 

.156302 

.93 

45 

.758481 

3.33 

.128492 

7.80 

45 

.770323 

3.25 

.156778 

.93 

46 

.758681 

3.32 

.128960 

7.78 

46 

.770518 

3.25 

.157254 

.93 

47 

.758880 

3.32 

.129427 

7.78 

47 

.770713 

3.27 

.157730 

.<« 

48 

.759079 

3.32 

.129894 

7.80 

48 

.770909 

3.25 

.158-206 

.95 

49 

.759278 

3.32 

.130362 

7.80 

49 

.771104 

3.25 

.15HC83 

M 

50 

.759477 

3.32 

.130830 

7.80 

50 

.771299 

3.25 

.159159 

.05 

51 

9.759676 

3.32 

10.131298 

7.80 

51 

9.771494 

3.25 

10.159636 

.95 

52 

.759875 

3.30 

.131766 

7.80 

52 

.771689 

3.25 

.160113 

.95 

53 

.760073 

3.32 

132234 

7.80 

53 

.771884 

3.25 

.160590 

.86 

54 

.760272 

3.32 

.132702 

7.80 

54 

.772079 

3.25 

.161067 

.1)7 

55 

.760471 

3.30 

.133170 

7.82 

55 

.772274 

3.25 

.161545 

.96 

56 

.760669 

3.32 

.133639" 

7.82 

56 

.772469 

3.25 

.162022 

.97 

57 

.760868 

3.30 

.134108 

7.82 

57 

.772664 

3.23 

.162500 

.97 

58 

.761066 

3.32 

.134577 

7.82 

58 

1779858 

iiss 

.162978 

.97 

59 

.761265 

3.30 

.ia5046 

7.H2 

59 

.773053 

3.25 

.163456 

7.97 

60 

9.761463 

3.30 

10.135515 

7.82 

60 

9.773248 

3.23 

10.163934 

7.98 

436 


AND  EXTERNAL  SECANTS. 


66° 

67° 

' 

Yers. 

D.  1".  Ex.  sec. 

D.  1'. 

* 

Yers. 

D.  1'. 

Ex.  sec. 

D.r. 

0 

9.773248 

8.23  10.103934 

7.98 

0 

9.784809 

3.18 

10.192931 

8.15 

i 

.  773442 

3.23 

.104413- 

7.97 

1 

.785000 

3.18 

.193420 

8.13 

2 

.773030 

3.25 

.104891 

7.98 

2 

.785191 

3.17 

.193908 

8.15 

3 

.773831 

3.23 

.105370 

7.98 

3 

.785381 

3.18 

.194397 

8.15 

4 

.774025 

3.23 

.105849 

7.98 

4 

.785572 

3.18 

.194886 

8.17 

5 

.774219 

3.25 

.100328 

7.98 

5 

.785763 

3.17 

.195376 

8.15 

6 

.774414 

3.23 

.100807 

7.98 

c 

.785953 

3.18 

.195865 

8.17 

7 

.774008 

3.23 

.107286 

8.00 

7 

.786144  3.17 

.196355 

8.17 

8 

.774803 

3.23 

.107700 

7.98 

8 

.780334 

3.17 

.196845 

8.17 

9  i  .774990 

3.23 

.108245 

8.00 

9 

.780524 

3.18 

.197335 

8.17 

10 

.775190 

3.23 

.108725 

8.00 

10 

.780715 

3.17 

.197825 

8.17 

It 

9.775384 

3.22 

10.109205 

8.00 

11 

9.78690') 

3.17 

10.198315 

8.18 

8 

.775577  1  3.23 

.109085 

8.00 

12 

.787C95 

3.17 

.198806 

8.18 

13 

.775771 

3.23 

.170105 

8.02 

13 

.787285 

3.17 

.199297 

8.18 

14 

.775905 

3.23 

.170040 

8.02 

14 

.787475 

3.17 

.199788 

8.18 

15 

.770159 

3.22 

.171127 

8.00 

15 

.787665 

3.17 

.200279 

8.18 

10 

.770352 

3.23 

.171007 

8.02 

16 

.787855 

3.17 

.200770 

8.20 

17 

.770540 

3.22 

.172088 

8.02 

17 

.788045 

3.17 

.201262 

8.18" 

18 

.770739 

3.23 

.172509 

8.03 

18 

.788235 

3.17 

.201753 

8.20 

19 

.770933 

3.22 

.173051 

8.02 

19 

.788425 

3.15 

.202245 

8.20 

20 

.777120 

3.22 

.173532 

8.03 

20 

.788614 

3.17 

.202737 

8.20 

21 

9.777319 

3.22 

10.174014 

8.03 

21 

9.788804 

3.15 

10.203229 

8.22 

22 

.777512 

3.22 

.174490 

8.03 

22 

.788993 

3.17 

.203722 

8.22 

23 

.777705 

3.23 

.174978 

8.03  |  23 

.789183 

3.15 

.204215 

8.20 

24 

.777899 

•3.22 

.175400 

8.03 

24 

.789372 

3.17 

.2J4707 

8.22 

25 

.778092 

3.22 

.175942 

8.05 

25 

.789502 

3.15 

.205200 

8.23 

20 

.778285 

3.20 

.170425 

8.03 

26 

.789751 

3.15 

.205694 

8.22 

27 

.778477 

3.22 

.170907 

8.05 

27 

.789940 

3.17 

.206187 

8.23 

28 

.778070 

3.22 

.177390 

8.05 

28 

.790130 

3.15 

.206081 

8.22 

29 

.778803 

3.22 

.177873 

8.05   29 

.790319 

3.15 

.207'!  74 

8.23 

30 

.779050 

3.20 

.178350 

8.05 

30 

.790508 

3.15 

.207068 

8.23 

31 

9.779248 

3.22. 

10.178839 

8.07 

31 

9.790097 

3.15 

10.208162 

8.25 

32 

.779441 

3.22 

.179323 

8.07 

32 

.790886 

3.15 

.208057 

8.23 

33 

.779034 

3.20 

.179807 

8.05 

33 

.791075 

3.15 

.209151 

8.25 

31 

.779820 

3.20 

.180290 

8.07 

34 

.791264 

3.15 

.209646 

8.25 

35 

.780018 

3.22 

.180774 

8.08 

35 

.791453 

3.13 

.210141 

8.25 

30 

.780211 

3.20 

.181259 

8.07 

36 

.791041 

3.15 

.210636 

8.25 

37 

.780403 

3.20 

.181743 

8.07 

37 

.791830 

3.15 

.211131 

8.27 

38 

.780595 

3.20 

.182227 

8.08 

38 

.792019 

3.13 

.211627 

8.27 

39 

.780787 

3.22 

.182712 

8.08 

39 

.792207 

3.15 

.212123 

8.25 

40 

.780980 

3.20 

.183197 

8.08 

40 

.792396 

3.13 

.212018 

8.28 

41 

9.781172 

3.20 

10.183082 

808 

41 

9.792584 

3.13 

10.213115 

8.27 

42 

.781304 

3.20 

.184107 

8.10 

42 

.792772 

3.15 

.  .213011 

8.27 

43 

.781550 

3.18 

.184053 

8.08 

43 

.792961 

3.13 

.214107 

8.28 

44 

.781747 

3.20 

.185138 

8.10 

44 

.793149 

3.13 

.214004 

8.28 

45 

.781939 

3.20 

.185024 

8.10 

45 

.793337 

3.13 

.215101 

8.28 

40 

.782131 

3.20 

'  .180110 

8.10 

46 

.793525 

3.15 

.215598 

8.28 

47 

.782323 

3.18 

.180596 

8.10 

47 

.793714 

3.13 

.210095 

8.30 

48 

.782514 

3.20 

.187082 

8.10 

48 

.793902 

3.13 

.216593 

8.28 

49 

.782700 

3.18 

.187568 

8.12 

49 

.794090 

3.12 

.217090 

8.30 

50 

.782897 

3.20 

.188055 

8.12 

50 

.794277 

3.13 

.217588 

8.30 

51 

9.  783089 

3.18 

10.188542 

8.12 

51 

9.794405 

3.13 

10.218086 

8.32 

52 

.783280 

3.18 

.189029 

8.12 

52 

.794053 

3.13 

.218585 

8.30 

53 

.783471 

3.20 

.189516 

8.13 

53 

.794841 

3.12 

.219083 

8.32 

54 

.783003 

3.18 

.190003 

8.13 

54 

.795028 

3.13 

.219582 

8.32 

55 

.7&S854 

3.18 

.190491 

8.12 

55 

.795216 

3.13 

.220081 

8.32 

50 

.784045 

3.18 

.190978 

8.13 

56 

.795404 

3.12 

.220580 

8.32 

57 

.784236 

3.18 

.191466 

8.13 

57 

.7'95591 

3.13 

.221079 

8.32 

58 

.784427 

3.18 

.191954 

'  8.15 

58 

.795779 

3.12 

.221578 

8.  S3 

59 

.784018 

3.18 

.192443 

8.13 

'59 

.795906 

3.12 

.222078 

8.33 

00 

9.784809  i  3.18 

10.192931 

8.15 

60 

9.796153 

3.13 

10.222578 

8.33 

437 


TABLE  XXVI. -LOGARITHMIC  VERSED  SINES 


r 

68° 

69° 

/ 

Vers. 

D.r. 

Ex.  sec. 

D.I'. 

/ 

Vers. 

D.  1".  Ex.  sec. 

D.r. 

0 

9.796153 

3.13 

10.222578 

8.33    0 

9.807286   3.07   10.252957 

8.55 

i 

.796341 

3.12 

.223078 

8.33  :   1 

.807470 

3.07    .253470 

8.55 

2 

.796528 

3.12 

.223578 

8.a5  i  2 

.807654' 

3.05 

.253983 

8.57 

3 

.796715 

3.12 

.224079 

8.33 

3 

.807837 

3.07    .254497 

8.55 

4 

.796902 

3.12 

.224579 

8.35 

4 

.808021 

3.05 

.255010 

8.57 

5 

.797089 

3.12 

.225080 

8.35 

5 

.808204 

3.07 

.255524 

8.58 

6 

.797276 

3.12 

.225581 

8.37 

6 

.808388  3.05 

.256039 

8.57 

7 

.797463 

3.12 

.226083 

8.35    7 

.808571   3.07 

.256553  !  8.58 

8 

.797650 

3.12 

.226584 

8.37    8 

.808755 

3.05 

.257068  !  8.57 

9 

.797837 

3.10 

.227086 

8.37    9 

.808938  3.05 

.257582  !  8.00 

10 

.798023 

3.12 

.227588 

8.37 

10 

.809121  3.07 

.258098  i  8.58 

j 

11 

9.798210 

3.12 

10.228090 

8.37 

11 

9.809305  3.05 

10.258613  ;  8.  CO 

12 

.798397 

3.10 

.228592 

8.38   12 

.809488  i  3.05 

.259129  1  8.58 

13 

.798583 

3.12 

.229095 

8.38   13 

.809671  !  3.05 

.259644   8.60 

14 

.798770 

3.10 

.229598 

8.38  i  14 

.809854 

3.05 

.260160 

8.62 

15 

.798056 

3.10 

.230101 

8.38  i  15 

.810037 

3.05 

.260677  8.60 

16 

.799142 

3.12 

.230604 

8.38  |  16   .810220 

3.05 

.261193  8.62 

17 

.799329 

3.10 

.231107 

8.40  i  17   .810403 

3.03 

.261710  8.62 

18 

.799515 

3.10 

.231611 

8.40  II  18 

.810585  3.05 

.262227  !  8.62 

19 

.799701 

3.10 

.232115 

8.40 

19 

.810768  3.05 

.262744 

8.63 

20 

.799887 

3.12 

.232619 

8.40 

20 

.810951   3.05 

.263262 

8.62 

21 

9.800074 

3.10 

10.233123 

8  40 

21 

9.811134  3.03 

10.263779 

8.63 

22 

.800260 

3.10 

.233627 

8.42 

22 

.811316  3.05 

.264297  8.63 

23 

.800446 

3.08 

.234132 

8.42 

23 

.811499  3.03 

.264815  8.65 

24 

.800631 

3.10 

.234637 

8.42   24 

.811681 

3.05 

.265334  8.65 

25 

.800817 

3.10 

.235142 

8.42  |  25 

.811864 

3.03 

.265853  8.63 

26 

.801003 

3.10 

.235647 

8.43   26 

.812046  3.03 

.266371  i  8.67 

27 

.801189 

3.10 

.236153 

8.42   27 

.812228  3.03    .266891   8.65 

28 

.801375 

3.08 

.236658 

8.43  !  28 

.812410 

3.05    .267410  8.67 

29 

.801560 

3.10 

.237164 

8.43 

29 

.812593 

3.03    .267930 

8.65 

30 

.801746 

3.08 

.237670 

8.45 

30 

.812775 

3.03    .268449 

8.68 

31 

9.801931 

3.10 

10.238177 

8.43 

31 

9.812957 

3.03  10.268970  8.67 

32 

.802117 

3.08 

.238683 

8.45 

32 

.813139 

3.03 

.269490  8.68 

33 

.802302 

3.08 

.239190 

8.45 

33 

.813321   3.03 

.270011 

18.67 

34 

.80248? 

3.10 

.239697 

8.45 

34 

.81:3503  3.03 

.270531 

8.68 

35 

.802673 

3.08 

.240204 

8.47 

35 

.813685  1  3.02 

.271052  i  8.70 

36 

.802858 

3.08 

.240712 

8.45 

36 

.813866 

3.03  !   .271574  8.68 

37 

.803043 

3.08 

.241219 

8.47 

37 

.814048 

3.03    .272095  8.70 

38 

.803228 

3.08 

.241727 

8.47 

38 

.814230 

3.02    .272617  ,  8.70 

39 

.803413 

3.08 

.242235 

8.48 

39 

.814411 

3.03 

.273139  8.72 

40 

.803598 

3.08 

.242744 

8.47 

40 

.814593 

3.03 

.273662  8.70 

41 

9.803783 

3.08 

10.243252 

8.48 

41 

9.814775 

3.02 

10.274184  8.72 

42 

.803968 

3.08 

.243761 

8.48 

42 

.814956 

3.02  i   .274707  8.72 

43 

.804153 

3.08 

.244270 

8.48 

43 

.815137 

3.03    .275230  8.72 

44 

.804338 

3.07 

.244779 

8.50 

44 

.815319   3.02    .275753  8.73 

45 

.804522 

3.08 

.245289 

8.48 

45 

.815500  3.02    .276277  8.73 

46 

.804707 

3.08 

.2457S8   8.50   46 

.815681  3.02    .276801  8.73 

47 

.804892 

3  07 

.246308   8.50 

47 

.815862  !  3.03 

.277325  8.73 

48 

.805076 

3.08 

.246818  i  8.52 

48 

.816044  3.02 

.277X19  8.75 

49 

.805261 

3.07 

.247329   8.50 

1  49 

.816225  3.02    .278374  8.15 

,50 

.805445 

3.07 

.247839 

8.52 

50 

.816406  3.02 

.278899 

8.75 

!51 

9.805629 

3.08 

10.248350 

8.52 

51 

9.816587 

3.00 

10.279424 

8.75 

52 

.805814 

3.07 

.248861  !  8.52 

52 

.816767 

3.02 

.279949  8.77 

53 

.805998 

3.07 

.249372  ]  8.52 

53 

.816948 

3.02 

.280475  '  8.75 

54 

.806182 

3.07 

.249883  i  8.53 

1  54 

.817129 

3.02 

.281000  i  8.78 

55 

.806366 

3.07 

.250395  i  8.53   55 

.817310 

3.00 

.281527  8.77 

56 

.806550 

3.07 

.250907  1  8.53   5(5 

.817490 

3.02 

.282053  |  8.78 

57 

.806734 

3.07 

.251419   8.55 

57 

.817671 

3.02 

.282580 

8.77 

58 

.806918 

3.07 

.251932   8.53 

58 

.817858 

3.00 

.283106 

8.80 

59 

.807102 

3.07 

.252444  !  8.55 

59 

.818032 

3.02 

.283634  8.78 

60  9.807286  3.07  10.252957  i  8.55 

GO  9.818213  i  3.00  10.284161  i  8.80 

AND  EXTERNAL  SECANTS 


70 

> 

' 

71° 

' 

Vers. 

D.  r. 

Ex.  sec. 

D.  r. 

'i 

Vers. 

D.I'. 

Ex.  sec. 

D.I'. 

0 

9.818213 

3.00 

10.284161 

8.80 

0 

9.828938 

2.95 

10.316296 

9.07 

1 

.818393 

3.00 

.284689 

8.78 

1 

.829115 

2.95 

.316840 

9.08 

2 

.818573 

3.02 

.285216 

8.82 

2 

.829292 

2.95 

.317385 

9.07 

3 

.818754 

3.00 

.285745 

8.80 

3 

.829469 

2.95 

.317929 

9.10 

4 

.818934 

3.00 

.286273 

8.82 

4 

.829646 

2.95 

.318475 

9.08 

5 

819114 

3.00 

.286802 

8.82 

5 

.829823 

2.95 

.319020 

9.08 

6 

.819294 

3.00 

.287331 

8.82 

6 

.830000 

2.95 

.319565 

9.10 

.819474 

3.00 

.287860 

8.82 

7 

.830177 

2.93 

.320111 

9.12 

8 

.819654 

3.00 

.288389 

8.83 

8 

.830&53 

2.95 

.320658 

9.10 

9 

.819834 

3.00 

.288919 

8.83 

9 

.830530 

2.93 

.321204 

9.12 

10 

.820014 

3.00 

.289449 

8.83 

10 

.830706 

2.95 

.321751 

9.12 

11 

9.820194 

3.00 

10.289979 

8.85 

11 

9.830883 

2.93 

10.322298 

9.12 

12 

.820374 

2.98 

.290*10 

8.85 

12 

.831059 

2.95 

.322845 

9.13 

13 

.820553 

3.00 

.291041 

8.85 

13 

.831236 

2.93 

.323393 

9.13 

14 

.820733 

3.00 

.291572 

8.85 

14 

.831412 

2.95 

.323941 

9.13 

15 

.820913 

2.98 

.292103 

8.87 

15 

.831589 

2.93 

.324489 

9.15 

16 

.821092 

3.00 

.292635 

8.85 

16 

.831765 

2.93 

.325038 

9.15 

17 

.821272 

2.98 

.293166 

8.87 

17 

.831941 

2.93 

.325587 

9.t5 

18 

.821451 

3.00 

.293698 

8.88 

18 

.832117 

2.93 

.326136 

9.17 

19 

.821631 

2.98 

.294231 

8.88 

19 

.832293 

2.93 

.326686 

9.15 

20 

.821810 

2.98 

.294764 

8.87 

20 

.832469 

2.93 

.327235 

9.18 

21 

9.821989 

2.98 

10.295296 

8.90 

21 

9.832645 

2.93 

10.327786 

9.17 

21 

.822168 

3.00 

.295830 

8.88 

22 

.832821 

2  93 

.328336 

9.18 

23 

.822348 

2.98 

.296363 

8.90 

23 

.832997 

2^93 

.328887 

9.18 

24 

.822527 

2.98 

.296897 

8.00 

24 

.833173 

2.93 

.329438 

9.18 

23 

.822706 

2.98 

.207431 

8.90 

25 

.833349 

2.93 

.329989 

9.20 

2(3 

.822885 

2.98 

.297965 

8.92 

26 

.833525 

2.92 

.330541 

9.20 

27 

.823064 

2.  98 

.298500 

8.90 

27 

.833700 

2.93 

.331093 

9.20 

23 

.823243 

2.07 

.299034 

8.93 

28 

.8:33876 

2.92 

.331645 

9.22 

29 

.823421 

2.98 

.299570 

8.92 

29 

834051 

2.93 

.a32198 

9.20 

30 

.823600 

2.98 

.300105 

8.93 

30 

.834227 

2.92 

.332750 

9.23 

31 

9.823779 

2.98 

10.300041 

8.92 

31 

9.834402 

2.93 

10.333304 

9.22 

32 

.823958 

2.97 

.301176 

8.95 

32 

.834578 

2.92 

.333857 

9.23 

33 

.824136 

2.98 

.301713 

8.93 

33 

.834753 

2.92 

.334411 

9.23 

34 

.824315 

2.97 

.302249 

8.95 

34 

.834928 

2.93 

.334965 

9.25 

35 

.824493 

2.98 

.302786 

8.95 

35 

.835104 

2.92 

.335520 

9.23 

30 

.824672 

2.97 

.303323 

8.95 

|  36 

.835279 

2.92 

.336074 

9.25 

37 

.824850 

2.97 

.303860 

8.97 

37 

.835454 

2.92 

.336629 

9.27 

38 

.8250.38 

2.98 

.304398 

8  97 

38 

.835629 

2.92 

.337185 

9.27 

33 

.825207 

2.97 

.304936 

8.97 

39 

.835804 

2.92 

.337741 

9.27 

40 

.825385 

2.97 

.305474 

8.97 

40 

.835979 

2.92 

.338297 

9.27 

41 

9.825563 

2.97 

10.306012 

8.98 

41 

9.836154 

2.92 

10.338853 

9.28 

42 

.825741 

2.97 

.306551 

8.98 

42 

.836329 

2.92 

.339410 

9.28 

43 

.825919 

2.97 

.307090 

8.98 

43 

.836504 

2.90 

.339967 

9.28 

44 

.826037 

2.97 

.307629 

9.00 

44 

.836678 

2.92 

.340524 

9.30 

45 

.826275 

2.97 

.308169 

8.98 

45 

.836853 

2.92 

.341082 

9.30 

46 

.826453 

2.97 

.308708 

9.02 

46 

.837028 

2.90 

.341640 

9.30 

47 

.826631 

2.97 

.309249 

9.00 

47 

.837202 

2.88 

.342198 

9.30 

48 

.826809 

2.97 

.309789 

9.02 

48 

.837377 

2.90 

.342756 

9.32 

49 

.826987 

2.95 

.310330 

9.02 

49 

.837551 

2.92 

.343315 

9.33 

50 

.827164 

2.97 

.310871 

9.02 

50 

.837726 

2.90 

.343875 

9.32 

51 

9.827342 

2.95 

10.311412 

9.02 

51 

9.837900 

2.92 

10.344434 

9.33 

52 

.827519 

2.97 

.311953 

9.03 

52 

.838075 

2.90 

.344994 

9.33 

53 

.827697 

2.95 

.312495 

9.03 

53 

.838249 

2.90 

.345554 

9.35 

54 

.827874 

2.97 

.313037 

9.05 

54 

.838423 

2.90 

.346115 

9.35 

55 

.828052 

2.95 

.313580 

9.03 

55 

.  £38597 

2.90 

.346676 

9.35 

56 

.828229 

2.95 

.314122 

9.05 

56 

.838771 

2.90 

.347237 

9.35 

57 

.828400 

2.97 

.314665 

9.07 

57 

.838945 

2.90 

.347798 

9.37 

58 

.828584 

2.95 

.315209 

9.05 

58 

.889119 

2.90 

.348360 

9.37 

59 

.823761 

2.95 

.315752 

9.07 

|  59 

.839293 

2.90 

.348922 

9.38 

60 

9.828938 

2.95 

10.316296 

9.07 

60 

9.839467 

2.90 

10.349485 

9.38 

489 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


72 

73° 

/ 

Vers. 

D.  1. 

Ex.  sec. 

D.r. 

Vers. 

D.  1". 

Ex.  sec. 

D.I". 

IT 

9.839467 

2.90 

10.349485 

9.38 

0 

9.849805 

2.85 

10.383870 

9.73 

i 

.839641 

2.90 

.350048 

9.38  ! 

1 

.849976 

2.85 

.384454 

9.73 

2 

.839815 

2.90 

.350611 

9.40 

2 

.850147 

2.83 

.385038 

9.75 

3 

.839989 

2.88 

.351175 

9.38 

3 

.850317' 

2.85 

.385623 

9.77 

4 

.840162 

2.90 

.351738 

9.42 

4 

.850488 

2.83 

.386209 

9.75 

5 

.840336 

2.90 

.352303 

9.40 

5 

.850658 

2.  £5 

.386794 

9.77 

0 

.840510 

2.88 

.352867 

9.42 

C 

.850829 

2.83 

.387380 

9.78 

7 

.840683 

2.90 

.353432 

9.42 

7 

.850999 

2.83 

.387967 

9.78 

8 

.840857 

2.88 

.353997 

9.43 

8 

.851169 

2.85 

.388554 

9.78 

9 

.841030 

2.90 

.354563 

9.43 

9 

.851340 

2.  £3 

.389141 

9.78 

10 

.841204 

2.88 

.355129 

9.43 

10 

.851510 

2.83 

.389728 

9.80 

11 

9.841377 

2.88 

10.355695 

9.43 

11 

9.851680 

2.83 

10.390316 

9.82 

12 

.841550 

2.88 

.356261 

9.45 

12 

.851*50 

2.83 

.390905 

9.80 

13 

.841723 

2.88 

.356828 

9.45 

13 

.852020 

2.83 

.391493 

9.82 

14 

.841896 

2.90 

.357395 

9.47 

14 

.852190 

2.83 

.392082 

9.83 

15 

.842070 

2.88 

.357963 

9.47 

15 

.852360 

2.83 

.392672 

9.83 

16 

.842243 

2.88 

.358531 

9.47 

16 

.852530 

2.83 

.393262 

9.83 

17 

.842416 

2.88 

.359099 

9.48 

17 

.852700 

2.83 

.393852 

9.85 

18 

.842589 

2.88 

.359668 

9.48 

18 

.852870 

2.83 

.394443 

9.85 

19 

.842762 

2.87 

.360237 

9.48 

19 

.853040 

2.82 

.3950:34 

9.85 

20 

.842934 

2.88 

.360806 

9.50 

20 

.853209 

2.83 

.395625 

9.87 

21 

9.843107 

2.88 

10.361376 

9.50 

21 

9.853379 

2.83 

10.396217 

9.87 

22 

.843280 

2.88 

.361946 

9.50 

22 

.853549 

2.82 

.396809 

9.88 

23 

.843453 

2.87 

.362516 

9.52 

23 

.853718 

2.83 

.397402 

9.88 

24 

.843625 

2.88 

.363087 

9.52 

24 

.853888 

2.82 

.397995 

9.90 

25 

.843798 

2.87 

.363658 

9.52 

25 

.854057 

2.83 

.398589 

9.88 

26 

.843970 

2.88 

.364229 

9.53 

26 

.854227 

2.82 

.399182 

9.92 

27 

.844143 

2.87 

.364801 

9.53 

27 

.854396 

2.82 

.399777 

9.90 

28 

.844315 

2.88 

.365373 

9.53 

28 

.854565 

2.83 

.400371 

9.92 

29 

.844488 

2.87 

.365945 

9.55 

29 

.854735 

2.82 

.400966 

9.93 

30 

.844660 

2.87 

.366518 

9.55 

30 

.854904 

2.82 

.401562 

9.93 

31 

9.844832 

2.87 

10.367'091 

9.57 

31 

9.855073 

2.82 

10.402158 

9.93 

32 

.845004 

2.88 

.367665 

9.57 

32 

.855242 

2.82 

.402754 

9.95 

33 

.845177 

2.87 

.368239 

9.57 

33 

.855411 

2.82 

.403351 

9.95 

34 

.845349 

2.87 

.368813 

9.57 

34 

.£55580 

2.82 

.403948 

9.95 

35 

.845521 

2.87 

.369387 

9.58 

35 

.855749 

2.82 

.404545 

9.97 

36 

.845693 

2.87 

.369962 

9.60 

36 

.855918 

2.82 

.405143 

9.98 

37 

.845865 

2.87 

.370538 

9.58 

37 

.856087 

2.80 

.405742 

9.97 

38 

.846037 

2.85 

.371113 

9.60 

38 

.856255 

2.82 

.400340 

9.98 

39 

.846208 

2.87 

.371689 

9.62 

39 

.856424 

2.82 

.406939 

10.00 

40 

.846380 

2.87 

.372266 

9.60 

40 

.856593 

2.82 

.407539 

10.00 

41 

9.846552 

2.87 

10.372842 

9.62 

41 

9.856762 

2.80 

10.408139 

10.00 

42 

.846724 

2.85 

.373419 

9.63 

42 

.856930 

2.82 

.408739 

10.02 

43 

.846895 

2.87 

.373997 

9.63 

43 

.857099 

2.80 

.40U340 

10.02 

44 

.847067 

2  85 

.374575 

9.63 

44 

.857267 

2.82 

.409941 

10.03 

45 

.847238 

2.87 

.375153 

9.03 

45 

.857436 

2.80 

.410543 

10.03 

46 

.847410 

2.85 

.375731 

9.65 

46 

.857604 

2.80 

.411145 

10.03 

47 

.847581 

2.87 

.376310 

9.67 

47 

.857772 

2.82 

.411747 

10.05 

48 

.847753 

2.85 

.376890 

9.65 

48 

.857941 

2.80 

.412350 

10.07 

49 

.847924 

2.85 

.377469 

9.67 

49 

.858109 

2.80 

.412954 

10.05 

50 

.848095 

2.87 

.378049 

9.68 

50 

.858277 

2.80 

.413557 

10.07 

51 

9.848267 

2.85 

10.378630 

9.67 

51 

9.858445 

2.80 

10.414161 

10.08 

52 

.848438 

2.85 

.378210 

9.70 

52 

.858613" 

2.80 

.414766 

10.08 

53 

.848609 

2.85 

.379792 

9.68  ' 

53 

.858781 

2.80 

.415371 

10.08 

£4 

.848780 

2.85 

.380373 

9.70 

54 

.858949 

2.80 

.415976 

10.10 

£5 

.848951 

2.85 

.380955 

9.70 

55 

.859117 

2.80 

.416582 

10.12 

56 

.849122 

2.85 

.381537 

9.72 

56 

.859285 

2.80 

.417189 

10.10 

57 

.849293 

2.85 

.382120 

9.72 

57 

.859453 

2.80 

.417795 

10.12 

58 

.849464 

2.83 

.382703 

9.72 

5* 

.£59621 

2.78 

.-118402 

10.13 

59 

.849634 

2.85 

.383286 

9.73 

59 

.859788 

2.80 

.419010 

10.13 

60 

9.849805 

2.85 

10.383870 

9.73 

68 

9.859956 

2.80 

10.419618 

10.13 

440. 


AND  EXTERNAL  SECANTS. 


?4o 

75° 

'  I     Vers. 

D.  r. 

Ex.  sec.      D.  1". 

' 

Vers. 

D.  1".    Ex.  sec. 

D.  1". 

0 

9.8J9056 

2.80 

10.419618      10.13    j     0 

9.869924 

2.75  ;  10.  456928 

10.60 

1 

.860124 

2.78          .420226      10.15  |l     1 

.870089 

2.73 

.457564 

10.62 

o 

.860291 

2.80 

.4208:35     10.17    i     2 

.870253 

2.75 

.458201 

10.63 

3 

.860459 

2.78 

.421445 

10.15 

3 

.870418 

2.73 

.458839 

10.62 

4 

.860626 

2.80 

.422054 

10.17 

4 

.870582     2.75 

.459476 

10.65 

5 

.860794 

2.7'8 

.422064 

10.18 

5 

.870747     2.73 

.460115 

10.65 

6 

.860961 

2.78 

.423275 

10.18 

6 

.870911      2.75 

.460754 

10.65 

7 

.861128 

2.80 

.423880 

10.20 

7 

.871076  1  2.73 

.461&93 

10.67 

8 

.861296 

2.78 

.424498 

10.20 

8 

.871240  |  2.7-3 

.462033 

10.67 

9 

.861403 

2.78 

.425110 

10.20 

9 

.871404     2.73 

.462673 

10.68 

10       .801030 

2.78 

.425722 

10.22 

10 

.871568 

2.73 

.463314 

10  70 

11  |  9.861797 

2.78 

10.420335 

10.22 

11 

9.871732 

2.73 

10.463956 

10.70 

12  i      .Hlil  '.Hi  t 

2.78 

.420948 

10.23 

12 

.871896 

2.73 

.464598 

10.70 

13  1     .802131 

2.78 

.427502 

10.23 

13 

.872060 

2.73 

.465240 

10.72 

14 

.802298 

2.78 

.428176 

10.23 

14 

.87'2224 

2.73 

465883 

10.73 

15 

.802465 

2.78 

.428790 

10.27 

15 

.872388 

2.73 

.466527 

10.73 

16 

.862632 

2.78 

.420406 

10.25 

10 

.872552 

2.73 

.467171 

10.73 

17 

.862799 

2.77 

.430021 

10.27 

17 

.872716 

2.73 

.467815 

10.75 

18 

.862965 

2.78 

.430037 

10.27 

18 

.872880 

2.72 

.468460 

10.77 

19 

.863132 

2.78 

.431253 

10.28 

1!) 

.873043 

2.73 

.469106 

10.77 

20 

.863299 

2.77 

.431870 

10.30 

20 

.873207 

2.73 

.469752 

10.77 

21 

9.8G3465 

2.78 

10.432488 

10.28 

21 

9.873371 

2.72 

10.470398 

10.78 

22 

.863632 

2.78 

.433105 

10.32 

22 

.87'3534 

2.73 

.471045 

10.80 

23 

.863799 

2.77 

.488784 

10.30 

23 

.873698 

2.72 

.471693 

10.80 

24 

.868986 

2.77 

.434342 

10.32 

24 

.873861 

2.73 

.472341 

10.82 

25 

.864131 

2.78 

.434961 

10.33 

25 

.874025 

2.72 

.472990 

10.82 

2(3 

.864298 

2.77 

.485681 

10.33 

26 

.874188 

2.72 

.473639 

10.83 

27 

.864464 

2.77 

.436201 

10.33 

27 

.874351 

2.73 

.474289 

10.83 

28 

.834630 

2.78 

.430821 

10.35 

28 

.874515 

2.72 

.474939 

10.85 

29 

.804797 

2.77 

.437442 

10.37 

29 

.874678 

2.72 

.475590 

10.87 

30 

.864363 

2.77 

.438004 

10.37 

30 

.874841 

2.72 

.476242 

10.85 

31 

9.8G5129 

2.77 

10.438686 

10.37 

31 

9.875004 

2.72 

10.476893 

10.88 

32 

.805295 

2.77 

.439308 

10.38 

32 

.875167 

2.72 

.477546 

10.88 

33 

.865461 

2.77 

.439931 

10.38 

33 

.875330 

2.72 

.478199 

10.88 

34 

.865627 

2.77 

.440654 

10.40 

34 

.875493 

2.72 

.47B852 

10.90 

35 

.865793 

2.77 

.441178 

10.40 

35 

.875656 

2.72 

.479506 

10.92 

36 

.865959 

2.75 

.441802 

10.42 

36 

.875819 

2.72 

.480161 

10.92 

37 

.866124 

2.77 

.442427 

10.42 

37 

.875982 

2.72 

.480816 

10.93 

38 

.866290 

2.77 

.443052 

10.43 

38 

.876145 

2.72 

.481472 

10.93 

39 

.866456 

2.77 

.443678 

10.43 

39 

.87'6308 

2.70 

.482128 

10.95 

40 

.806022 

2.75 

.444304 

10.45 

40 

.876470 

2.72 

.482785 

10.05 

41 

9.866787 

2.77 

10.444931 

10.45 

41 

9.876633 

2.72 

10.483442 

10.97 

42 

.8(ir><)53      2.75 

.445558 

10.45 

42 

.876796 

2.70 

.484100 

10.1)8 

43 

.Si  17118 

2.77 

.446185 

10.47 

43 

.876958 

2.72 

•     .484759 

10.98 

44 

.867284 

2.75 

.446813 

10.48 

44 

.877121 

2.70 

.485418 

10.JJ8 

45 

.867449 

2.75 

.447442 

10.48 

45 

.877283 

2.70 

.486077 

10.1)8 

46 

.8i;rc,i  i 

2.77 

.448071 

10.48 

46 

.877445 

2.72 

.486738 

11.00 

47 

.867780 

2.75 

.448700 

10.50 

47 

.877608 

2.70 

.487398 

11  02 

48 

.867945 

2.75 

.449*30 

10.52 

48 

.877770 

2.70 

.488059 

11.03 

49 

.808110 

2.75 

.449961 

10.52 

j  49 

.877932 

2.7*2 

.488721 

11.05 

50 

.808275 

2.77 

.450592 

10.52 

50 

.878095 

2.70 

.489384 

11.05 

51 

9.868441 

2.75 

10.451223 

10.53 

51 

9.878257 

2.70 

10.490047 

11.05 

52 

.868606 

2.75 

.451855 

10.53 

52 

.878419 

2.70 

.490710 

11.07 

53 

.868771 

2.75 

.452487 

10.55 

53 

.878581 

2.70 

.491374 

11.08 

54 

.868936 

2.73 

.453120  I  10.57 

54 

.878743 

2.70 

.492039 

11.08 

55 

.809100 

2.75 

.453754 

10.57 

55 

.878905 

2.70 

.4927'04 

11.10 

56 

.869205 

2.75 

.454888 

10.57 

56 

.879067 

2.70 

.493370 

11.10 

57 

.889430 

2.75 

.4551)22 

10.58 

57 

.879229 

2.68 

.494036 

11.12 

58 

.869595 

2.75 

i'46MB7 

10.58 

58 

.879390 

2.70 

.494703 

11.13 

59 

.809760 

2.73 

.    .450292 

10.60 

59 

.879552 

2.70 

.495371 

11.13 

60 

9.809924 

2.75 

10.456928 

10.00 

60 

9.879714 

2.70 

10.496039 

11.18 

441 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


76 

o- 

77° 

f 

Vers. 

D.  r. 

Ex.  sec. 

D.  r. 

, 

Vers. 

D.r. 

Ex.  sec. 

D.  1". 

| 

Q 

9.879714 

2.70 

10.496039 

11.13 

0 

9.889329 

2.65 

10.537241 

11.77 

1 

.879876 

2.68 

.496707 

11.17 

1 

.889488 

2.65 

.537947 

11.78 

2 

.880037 

2.70 

.497377 

11.17 

2 

.889647 

2.63 

.538654 

11.80 

3 

.880199 

2.68 

.498047 

11.17 

3 

.889805 

2.65 

.539362 

11.82 

4 

.880360 

2.70 

.498717 

11.18  | 

4 

.889964 

2.65 

.540071 

11.82 

5 

.8805-22 

2.68 

.499388 

11.20 

5 

.890123 

2.63 

.540780 

11.83 

6 

.880633 

2.70 

.500380 

11.20 

6 

.890281 

2.65 

.541490 

11.83 

7 

.880345 

2.68 

.500732 

11.22 

7 

.890440 

.2.63 

.542200 

11.85 

8 

.881006 

2.68 

.501405 

11.22 

8 

.890598 

2.65 

.542911 

11.87 

9 

.881167 

2.70 

.502078 

11.23 

9 

.890757 

2.63 

.543623 

11.88 

10 

.881329 

2.68 

.502752 

11.23 

10 

.890915 

2.63 

.544336 

11.88 

11 

9.881490 

2.68 

10.503423 

11.27 

11 

9.891073 

2.65 

10.545049 

11.90 

12 

.881651 

2.68 

.504102 

11.25 

12 

.891232 

2.63 

.545763 

11.90 

13 

.881812 

2.68 

.504777 

11.23 

13 

.891390 

2.63 

.546477 

11.93 

14 

.881973 

2.63 

.505454 

11.23  | 

14 

.891548 

2.63 

.547193 

11.93 

15 

.882134 

2.63 

.503131 

11.23  ! 

15 

.891706 

ST.  63 

.547909 

11.95 

16 

.882295 

2.68 

.505303 

11.30 

16 

.891864 

2.63 

.548626 

11.95 

17 

.882456 

2.63 

.507486 

11.32 

17 

.892022 

2.63 

.549343 

11.97 

18 

.882517 

2.67 

.508165 

11.32 

18 

.892180 

2.63 

.550061 

11.93 

19 

.882777 

2.63 

.503344 

11.33 

19 

.892338 

2.63 

.550780 

12.00 

20 

.882933 

2.68 

.509524 

11.35 

20 

.892496 

2.63 

.551500 

12.00 

21 

9.833039 

2.63 

10.510205 

11.35 

21 

9.892654 

2.63 

10.552220 

12.02 

22 

.833230 

2.67 

.510383 

11.37 

22 

.892312 

2.62 

.552941 

12.03 

23 

.833120 

2.68 

.511568 

11.37 

23 

.892989 

2.63 

.553663 

12.03 

2t 

.833531 

2.67 

.512253 

11.33 

24 

.893127 

2.63 

.554385 

12.07 

25 

.883741 

2.63 

.512933 

11.40 

25 

.893235 

2.62 

.555109 

12.07 

23 

.83*902 

2.67 

.513617 

11.40 

28 

.893142 

2.63 

.555833 

12.07 

27 

.834032 

2.63 

.514301 

11.42 

27 

.893600 

2.63 

.556557 

12.10 

28 

.834223 

2.67 

.514986 

11.43 

23 

.893758 

2.62 

.557283 

12.10 

29 

.884383 

2.67 

.515672 

11.43 

29 

.893915 

2.62 

.558009 

12.12 

30 

.884543 

2.6Z 

.516358 

11.45 

30 

.894072 

2.63 

.558736 

12.12 

31 

9.884703 

2.63 

10.517015 

11.45  i 

31 

9.894230 

2.62 

10.559463 

12.15 

33 

.884864 

2.67 

.51773-2 

11.47  I 

32 

.894337 

2.62 

.560192 

12.15 

33 

.885024 

2.67 

.518420 

11.48 

33 

.894544 

2.63 

.560921 

12.17 

34 

.885184 

2.67 

.519109 

11.48 

34 

.894702 

2.62 

.561651 

12.17 

35 

.885344 

2.67 

.519793 

11.50 

35 

.894859 

2.62 

.562381 

12.20 

36 

.885504 

2.67 

.520433 

11.52 

36 

.89.5016 

2.62 

.563113 

12.20 

37 

.835364 

2.67 

.521179 

11.50 

37 

.895173 

2.62 

.563845 

12.20 

38 

.885824 

2.65 

.5:21870 

11.53 

38 

.895330 

2.62 

.564577 

12.23 

39 

.885983 

2.67 

.522552 

11.53 

39 

.895487 

2.62 

.565311 

12.23 

40 

.886143 

2.67 

.523:254 

11.55 

40 

.895644 

2.62 

.566045 

12.27 

41 

9.886303 

2.67 

10.523947 

11.57 

41 

9.895801 

2.62 

10.566781 

12.25 

42 

.886463 

2.65 

.5-21641 

11.57 

42 

.895958 

2.62 

.567516 

12.28 

43 

.886622 

2.67 

.525335 

11.58 

43 

.896115 

2.62 

.568253 

12.28 

41 

.886782 

2.65 

.526030 

11.60 

44 

.896272 

2.60 

.568990 

12.32 

45 

.886941 

2.67 

.526726 

11.62 

45 

.896428 

2.62 

.569729 

12.32 

46 

.887101 

2.65 

.527423 

11.62 

46 

.896585 

2.62 

.570468 

12.32 

47 

.887260 

2.67 

.528120 

11.62 

47 

.896742 

2.60 

.571207 

12.35 

48 

.887420 

2.65 

.528817 

11.65 

48 

.896898 

2.62 

.571948 

12.35 

49 

.887579 

2.67 

.529316 

11.65 

49 

.897055 

2.60 

.572(5.89 

12.37 

50 

.887739 

2.65 

.530215 

11.65 

50 

.897211 

2.62 

.573431 

12.38 

51 

9.887898 

2.65 

10.530914 

11.67 

•51 

9.897368 

2.60 

10.574174 

12.38 

52 

.888057 

2.65 

.531614 

11.68 

52 

.897524 

2.60 

.574917 

12.42 

53 

.888216 

2.65 

.532315 

11.70 

53 

.897680 

2.62 

.575662 

12.42 

54 

.888375 

2.65 

.533017 

11.70 

54 

.897837 

2.60 

.576407 

12.43 

55 

.888534 

2.65 

.533719 

11.72 

55 

.897993 

2.  GO 

.577153 

12.45 

56 

888693 

2.65 

.534422 

11.73 

50 

.898149 

2.60 

.577900 

12.45 

57 

.888852 

2.65 

.535126 

11.73 

57 

.898305 

2.60 

.578647 

12.48 

58 

.889011 

2.65 

.535830 

11.75 

58 

.898461 

2.62 

.579396 

12.48 

59 

889170 

2.65 

.5315535 

11.77 

59 

.898618 

2.60 

.580145 

12.50 

60 

9.889329 

2.65 

10.537241 

11.77 

60 

9.898774 

2.60 

10.580895 

12.50 

443 


AND  EXTERNAL  SECANTS. 


78° 

79° 

/ 

Vers. 

D.  r. 

Ex.  sec. 

D.  r. 

/ 

Vers. 

D.I". 

Ex.  sec. 

D.  1'. 

0  !  9.898774 

2.60 

10.580895 

12.50  \\  0 

9.908051 

2.55 

10.627452 

13.40 

1 

.898930 

2.(iO 

.581645 

12.53    1 

.908204 

2.55 

.628256 

13.40 

2 

.899086 

2.58 

.582397 

12.  53 

2 

.908357 

2  57 

'  .629060 

13.43 

3 

.899241 

2.60 

.583149 

12.57 

3 

.908511 

2^55 

.629866 

13.45 

4 

.899397  2.60 

.583903 

12.57 

4 

.908664 

2.55 

.630673 

13.45 

5 

.899553 

2.60 

.584657 

12.57 

5 

.908817 

2.55 

.631480 

13.48 

G 

.899709 

2.60 

.585411 

12.60 

i  G 

.908970 

2.55 

.632289 

13.48 

7 

.899865  2.58 

.586167 

12.60 

7 

.909123 

2.55 

.633098 

13.52 

8 

.900020  2.60 

.586923 

12.63 

8 

.909276 

2.53 

.6&3909 

13.52 

9 

.900176 

2.58 

.587681 

12.63 

9 

.909428 

2.55 

.634720 

13.55 

10 

.900331 

2.60 

.588439 

12.65 

10 

.909581 

2.55 

.635533 

13.55 

11 

9.900487 

2.58 

10.589198 

12.65 

11 

9.909734 

2.55 

10.636346 

13.58 

12 

.900642 

2.60 

.589957 

12.68 

12 

.909887- 

2.53 

.637161 

13.58 

13 

.900798 

2.58 

.590718 

12.68 

13 

.910039 

2.55 

.637976 

13.60 

14 

.900953 

2.58 

.591479 

12.72 

14 

.910192 

2.55 

.638792 

13.63 

15 

.9,)1108 

2.60 

.592242 

12.72 

15 

.910345 

2.53 

.639610 

13.63 

1(5 

.901264 

2.58 

.593005 

12.73 

16 

.910497 

2.55 

.640428 

13.67 

17 

.901419 

2.58 

.593769 

12.73 

17 

.910650 

2.53 

.641248 

13.67 

18 

.901574 

2.58 

.594533 

12.77 

18 

.910802 

2.55 

.642068 

13.70 

19  !  .901729 

2.58 

.595299 

12.78 

19 

.910955 

2.53 

.642890 

13.72 

20 

.901884 

2.60 

.596066 

12.78 

20 

.911107 

2.53 

.643713 

13.72 

21 

9.902040 

2.58 

10.596833 

12.80 

21 

9.911259 

2.55 

10.644536 

13.75 

22 

.902195 

2.58 

.597601 

12.82 

22 

.911412 

2.53 

.645361 

13.75 

23 

.902350 

2  57 

.598370 

12.83 

23 

.911564 

2.53 

.646186 

13.78 

24 

.902504 

2.58 

.599140 

12.85 

24 

.911716 

2.53 

.647013 

13.80 

25 

.902659 

2.58 

.599911 

12.85 

25 

.911868 

2.53 

.647341 

13.82 

26 

.902814 

2.58 

.600682 

12.88 

26 

.912020 

2.53 

.648670 

13.82 

27 

.902969 

2.58 

.60M55 

12.88 

27 

.912172 

2.53 

.649499 

13.85 

28 

.903124 

2.57 

.602228 

12.92 

28 

.912324 

2.53 

.650330 

13.87 

29 

.903278 

2.58 

.603003 

12.92 

29 

.912476 

2.53 

.651162 

13.88 

30 

.903433 

2.58 

.603778 

12.93 

30 

.912628 

2.53 

.651995 

13.90 

31 

9.903588 

2.57 

10.604554 

12.95 

31 

9.912780 

2.53 

10.652829 

13.92 

32 

.903742 

2.58 

.605331 

12.95 

32 

.912932 

2.53 

.653664 

13.95 

33 

.903897 

2.57 

.606108 

12.98 

33 

.913084 

2.52 

.654501 

13.95 

34 

.904051 

2.58 

.606887 

13.00 

34 

.913235 

2.53 

.655338 

13.97 

35 

.904206 

2.57 

.607667 

13.00 

35 

.913387 

2.53 

.656176 

14.  CO 

36 

.904360 

2.57 

.608447 

13.02 

36 

.913539 

2.52 

.657016 

14.  CO 

37 

.904514 

2.57 

.609228 

13.03 

37 

.913690 

2.53 

.657856 

14.03 

38 

.904668 

2.58 

.610010 

13.07 

38 

.913842 

2.52 

.658698 

14.03 

39 

.904823 

2.57 

.610794 

1307 

39 

.913993 

2.53 

.659540 

14.07 

40 

.90J977 

2.57 

.611578 

13.08 

40 

.914145 

2.52 

.660384 

14.08 

41 

9.905131 

2.57 

10.612363 

13.08 

41 

9.914296 

2.53 

10.661229 

14.10 

42 

.905285 

2.57 

.613148 

13.12 

42 

.914448 

2.52 

.662075 

14.12 

43 

.905439 

2.57 

.613935 

13.13 

43 

.914599 

2.52 

.662922 

14.13 

44 

.905593 

2.57 

.614723 

13.13 

44 

.914750 

2.53 

.663770 

14.15 

45 

.905747 

2.57 

.615511 

13.17 

45 

.914902 

2.52 

.664619 

14.18 

46 

.905901 

2.57 

.616301 

13.17 

46 

.915053 

2.52 

.665470 

14.18 

47 

.906055 

2.57 

.617091 

13.20 

47 

.915204 

2.52 

.666321 

14.22 

48 

.906209 

2.57 

.617883 

13.20 

48 

.915355 

2.52 

.667174 

14.23 

49 

.906363 

2.55 

.618675 

13.22 

49 

.915506 

2.52 

.668028 

14.25 

50 

.906516 

2.57 

.619468 

13.23 

50 

.915657 

2.52 

.668883 

14.27 

51 

9.906670 

2.57 

10.620262 

13.25 

51 

9.915808 

2.52 

10.669739 

14.28 

52 

.906824 

2.55 

.621057 

13.27 

52 

.915959 

2.52 

.670596 

14.30 

53 

.906977 

2.57 

.621853 

13.28 

53 

.916110 

2.52 

.671454 

14.33 

54 

.907131 

2.55 

.622650 

13.30 

54 

.916261 

2.52 

.672314 

14.33 

55 

.907284 

2.57 

.623448 

13.32 

55 

.916412 

2.50 

.673174 

14.37 

56 

.907438 

2.55 

.624247 

13.33 

56 

.916562 

2.52 

.674036 

14.38 

57 

.907591 

2.55 

.625047 

13.35 

57 

.916713 

2.52 

.674899 

14.40 

58 

.907744  2.57 

.625848 

13.37 

58 

.9168C4 

2.50 

.675763 

14.42 

59 

.907898  2.55 

.626650 

13.37 

59 

.917014 

2.52 

.676628 

14.45 

60 

9.908051   2.55 

10.627452 

13.40 

60 

9.917165 

2.52 

10.677495 

14.45 

443  _ 


TABLE  XXVI.-LOGARITHMIC  VERSED  SINES 


80 

3 

I 

81° 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.  1", 

/ 

Vers. 

D.  1". 

Ex.  sec. 

D.I'. 

0 

9.917165 

2.52 

10.677495 

14.45 

0 

9.926119 

2.47 

10.731786 

15.78 

.917316 

2.50 

.678362 

14.48 

1 

.926267 

2.47 

.732733 

15.78 

2 

.917466 

2.50 

.679231 

14.50 

2 

.926415 

2.45 

.788680 

15.83 

3 

.917616 

2.52 

.680101 

14.52 

3 

.926562 

2.47 

.734630 

15.83 

4 

.917767 

2.50 

.680972 

14.55 

4 

.926710 

2.47 

.735580 

15.87 

5 

.917917 

2.52 

.681845 

14.55 

5 

.926858 

2.47 

.736532 

15.90 

6 

.918068 

2.50 

.682718 

14.58 

6 

.927006 

2.45 

.737486 

15.92 

7 

.918218 

2.50 

.683593 

14.60 

7 

.927153 

2.47 

.738441 

15.95 

8 

.918368 

2.50 

.684469 

14.62 

8 

.927301 

2.45 

.739398 

15.97 

9 

.918518 

2.50 

.685346 

14.63 

9 

.927448 

2.47 

.740356 

16.00 

10 

.918668 

2.50 

.686224 

14.67 

10 

.927596 

2.45 

.741316 

16.02 

11 

9.918818 

2.50 

10.687104 

14.68 

11 

9.927743 

2.47 

10.742277 

16.03 

12 

.918968 

2.50 

-.687985 

14.70 

12 

.927891 

2.45 

.743239 

16.08 

13 

.919118 

2.50 

.688867 

14.72 

13 

.928038 

2.45 

.744204 

16.08 

14 

.919268 

2.50 

.689750 

14.73 

14 

.92S185 

2.47 

.745109 

16.13 

15 

.919418 

2.50 

.690634 

14.77 

15 

.928333 

2-.  45 

.740137 

10.13 

16 

.919568 

2.50 

.691520 

14.78 

16 

.928480 

2.45 

.747105 

10.  IK 

17 

.919718 

2.50 

.692407 

14.80 

17 

.928627 

2.45 

.748076 

16.20 

18 

.919868 

2.50 

.693295 

14.83 

18 

.928774 

2.45 

.749048 

16.22 

19 

.920018 

2.48 

.694185 

14.83 

19 

.928921 

2.45 

.750021 

16.25 

20 

.920167 

2.50 

.695075 

14.87 

20 

.929068 

2.45 

.750996 

10  .£8 

21 

9.920317 

2.48 

10.695967 

14.90 

21 

9.929215 

2.45 

10.751973 

16.  SO 

22 

.920466 

2.50 

.690861 

14.90 

22 

.929868 

2.45 

.752951 

10.33 

23 

.920616 

2./>0 

.697755 

14.93 

23 

.929509 

2.45 

.753931 

10.35 

24 

.920766 

2.48 

.698651 

14.95 

24 

.929656 

2.45 

.7'54912 

10.38 

23 

.920915 

2.48 

.699548 

14.97 

25 

.929803 

2.45 

.755895 

16.42 

26 

.921064 

2.50 

.700446 

15.00 

26 

.929950 

2.45 

.756880 

10.43 

27 

.921214 

2.48 

.701346 

15.02 

27 

.930097 

2.43 

.757866 

16.47 

28 

.921363 

2.48 

.702247 

15.03 

28 

.930243 

2.45 

.758854 

16.50 

29 

.921512 

2.50 

.703149 

15.05 

29 

.930390 

2.45 

.759844 

10.52 

30 

.921662 

2.48 

.704052 

15.08 

30 

.930537 

2.43 

.760835 

10.53 

31 

9.921811 

2.48 

10.704957 

15.10 

31 

9.930683 

2.45 

10.761827 

16.58 

32 

.921960 

2.48 

.705863 

15.13 

32 

.930830 

2.43 

.762822 

10.  GO 

33 

.92-2109 

2.48 

.706771 

15.15 

33 

.930976 

2.45 

.763818 

1C.C2 

34 

.922258 

2.48 

.707680 

15.17 

34 

.931123 

2.43 

.764815 

10.07 

35 

.922407 

2.48 

.708590 

15.18 

35 

.931269 

2.45 

.765815 

10.08 

3(3 

.922556 

2.48 

.709501 

15.22 

36 

.931416 

2.43 

.766816 

16  .TO 

37 

.922705 

2.48 

.710414 

15.23 

37 

.931562 

2.43 

.767819 

16.73 

38 

.922854 

2.48 

.711328 

15.25 

38 

.931708 

2.45 

.768823 

16.77 

89 

.923003 

2.48 

.712243 

15.28 

39 

.931855 

2.43 

.769859 

10.80 

40 

.923152 

2.48 

.713160 

15.30 

40 

.932001 

2.43 

.770837 

10.82 

41 

9.923301 

2.47 

10.714078 

15.33 

41 

9.932147 

2.43 

10.771840 

10.87 

42 

.923449 

2.48 

.714998 

15.35 

42 

.932293 

2.43 

.7'7'2t5S 

lelg? 

43 

.923593 

2.48 

.715919 

15.37 

43 

.932439 

2.43 

[773870 

16.  £2 

44 

.923747 

2.47 

.716841 

15  £8 

44 

.932585 

2.43 

.774885 

1C.  85 

45 

.923895 

2.48 

.717764 

15.42 

45 

.032731 

2.43 

.775902 

10.1-7 

46 

.924044 

2.47 

.718689 

15.45 

46 

.932877 

2.43 

.770920 

1  .CO 

47 

.924192 

2.48 

.719616 

15.45 

47 

.933023 

2.43 

.777940 

1  .02 

48 

.924241 

2.47 

.720543 

15.48 

48 

.933169 

2.43 

.77'8961 

1  .07 

49 

.904489 

2.47 

.121472 

15.52 

49 

.983315 

2.42 

.779985 

:  .08 

50 

.924637 

2.48 

.722403 

15.53 

50 

.933460 

2.43 

.781010 

1  .12 

51 

9.9247B6 

2.47 

10.72J3335 

15.55 

51 

9.933006 

2.43 

10.782037 

1  .13 

ca 

.924934 

2.47 

.7242(58 

15.58 

52 

.933752 

2.42 

.78COC5 

1  .18 

53 

.925082 

2.48 

.725203 

15.  GO 

53 

.933897 

2.43 

.784096 

1  .20 

54 

.925231 

2.47 

.726139 

15.63 

54 

.934043 

2.43 

.783128 

1  .2.'} 

55 

.925379 

2.47 

.727077 

15.65 

55 

.934189 

2.42 

.786162 

1  .27 

16 

.925527 

2.47 

.728016 

15.07 

66 

.9:34334 

2  43 

.787198 

1  .30 

57 

.925075 

2.47 

.728956 

15.70 

57 

.934480 

2.42 

.788236 

I  .33 

58 

.925823 

2.47 

.729898 

15.73 

58 

.934625 

2.42 

.7'89276 

1  .35 

59 

.925971 

2.47 

.730842 

15.73 

59 

.934770 

2.43 

.790317 

1  .40 

CO 

9.926119 

2.47 

10.731786 

15.78 

60 

9.934916 

2.42 

10.791361 

1  .42 

444 


AND  EXTERNAL  SECANTS. 


82° 

83° 

Vers. 

D.  1'.  Ex.  sec. 

D.  1". 

, 

Vers. 

D.  1".  Ex.  sec. 

D.I*. 

1 

I 

0 

9.934916 

2.42 

10.791301 

17.42 

o 

9.943559 

2.38  10.857665 

19.55 

1 

.9:55061 

2.42 

.792406 

17.45 

1 

.943702 

2.38   .858838 

19.58 

2 

.935206 

2.43 

.793453 

17.48 

2 

.943845 

2.37 

.860013 

19.63 

3 

.935352 

2.42 

,794502 

17.50 

3 

.943987 

2.38 

.861191 

19.67 

4 

.935497 

2.42 

.795552 

17.55 

4 

.944130 

2.38 

.862371 

19.72 

5 

.935642 

2.42 

.796605 

17.58 

5 

.944273 

2.37 

.863554 

19.75 

6 

.935787 

2.42 

.797680 

17.60 

6 

.944415 

2.38 

.864739 

19.80 

7 

.935932 

2.42 

.798716 

17.63 

7 

.944558  i  2.37 

.865927 

19.83 

8 

.936077 

2.42 

.799774 

17.68 

8 

.944700  2.38 

.867117 

19.88 

9 

.936222 

2.42 

.800835 

17.70 

1  9 

.944843  1  2.37 

.868310 

19.92 

10 

.936367 

2.42 

.801897 

17.73 

!  10 

.944985  2.37 

.869505 

19.97 

11 

9.936512 

2.42 

10.802961 

17.77 

11 

9.945127  2.38 

10.870703 

20.00 

19 

.936657 

2.40 

.804027 

17.80   12 

.945270  !  2.37 

.871903 

20.05 

18 

.936801 

2.42 

.805095 

17.83 

1  13 

.945412  i  2.37 

.873106 

20.10 

11 

.936946 

2.42 

.806165 

17.87 

14 

.945554  j  2.37 

.874312 

20.13 

15 

.937091 

2.42 

.807237 

17.90 

1  15 

.945696  1  2.37 

.875520 

20.18 

16 

.937236 

2.40 

.808311 

17.93 

16 

.945838  i  2.38 

.876731  '  20.23 

17 

.937380 

2.42 

.809387 

17.97 

17 

.945981   2.37 

.877945  !  20.27 

18 

.937525 

2.40 

.810465 

18.00 

1  18 

.946123  2.37 

.879161  i  20.30 

11) 

.937669 

2.42 

.811545 

18.03 

19 

.946265 

2.37 

.880379 

20.37 

20 

.937814 

2.40 

.812627 

18.07 

20 

.946407 

2.37 

.881601 

20.40 

21 

9.937958 

2.42 

10.813711 

18.10 

21 

9.946549 

2.35 

10.882825 

20.45 

23 

.938103 

2.40 

.814797 

18.13 

22 

.946690 

2.37 

.884052 

20.48 

23 

.938847 

2.40 

.815885 

18.17 

.946832 

2.37 

.885281 

20.55 

24 

.938391 

2.42 

.816975 

18.20 

24 

.946974 

2.37 

.886514 

20.58 

25 

.938536 

2.40 

.818067 

18.23 

25 

.947116 

2.37 

.887749 

20.62 

20 

.938680 

2.40 

.819161   18.27 

i  26 

.947258 

2.35 

.888986 

20.68 

27 

.938824 

2.40 

.820257  18.32 

!  27 

.947399 

2.37 

.8905227 

20.72 

28 

.938968 

2.40 

.821356  18.  as  1  28 

.947541 

2.37 

.891470 

20.77 

20 

.939112  2.38 

.822456  i  18.38  ji  29 

.947683 

2.35 

.892716 

20.82 

30 

.939257 

2.40 

.823559  i  18.42 

30 

.947824 

2.37 

.893965 

20.87 

31 

9.939101 

2.40 

10.824664  18.43 

81 

9.947966 

2.35 

10.895217 

20.92 

32 

.939545 

2.38    .825770  18.48 

32 

.948107 

2.37 

.896472 

20.95 

33 

9396.88 

2.40 

.826879  !  18.52 

33 

.948249 

2.35 

.897729 

21.00 

34 

.  939832 

2.40 

.827990  ;  18.57 

34 

.948390 

2.35 

.898989 

21.07 

35 

!  939976  !  2.40 

.829104  18.58 

35 

.948531 

2.37 

.900253  21.10 

36 

.940120  i  2.40 

.830219  18.63 

36 

.948673 

2.35 

.901519 

21.15 

37 

.940264  !  2.40 

.831337  18.65 

37 

.948814 

2..  35 

.902788 

21.20 

3S 

.940408  |  2.38 

.83-3456 

18.70 

38 

.948955 

2.35 

.904060 

21.25 

39 

.940551  (2.40 

.a33578 

18.75 

39 

.949096 

2  -.35 

.9053:35 

21.30 

40 

.940695 

2.40 

.834703 

18.77 

40 

.949237 

2.37 

.906613 

21.33 

41 

9.910839 

2.38 

10.8.35829 

18.80 

41 

9.949379 

2.35 

10.907893 

21.40 

42 

.940982 

2.40 

.836957 

18.85 

42 

.949520 

2.35 

.909177 

21.45 

43 

.911126 

2.38 

.&38088 

18.88 

43 

.949661 

2.35 

.910464 

21.50 

44 

.941369 

2.40 

.839221 

18.93 

44 

.949802 

2.35 

.911754 

21.55 

45 

.911413 

2.38 

.840357 

18.95 

45 

.949943 

2.33 

.913047 

21.60 

46 

.9115513 

2.38 

.841494 

19.00 

46 

.950083 

2.35 

.914343 

21.65 

47 

.941699 

2.40 

.842634 

19.03 

47 

.950224 

2.35 

.915642 

21.70 

48 

.941843 

2.38 

.843778 

19.08 

48 

.950365 

2.35 

.916944 

21.75 

49 

.941936  !  2.38 

.841921 

19.12 

49 

.950306 

2.35 

.918249  !  21.82 

50   .942129  2.33 

.8460G8 

19.15 

50   .95C647 

2.33 

.919558  21.85 

51   9.942272  2.38 

10.847217 

19.18 

51 

9.950787 

2  35  10.920869  21.92 

52 

.942415  2.40 

.848368 

19.23   52 

.950928 

2.35 

.922184  21.97 

53 

.942559  2.38 

.849522 

19.27 

53 

.951069 

2.33 

.923502  22.02 

54 

.942702  i  2.38 

.850678 

19.30  i 

54 

.951209 

2.35 

.924823  j  22.07 

55 

.912845  i  2.38 

.851836 

19.35 

55 

.951350 

2.33 

.926147 

22.13 

56 

.942988  \  2.38 

.852997 

19.40  i 

56 

.951,490 

2.35 

.927475 

22.17 

57 

.943131  i  2.37 

.854161 

19.42  ' 

57 

.951631 

2.33 

.928805 

22.23 

58 

.943273  !  2.38 

.855326 

19.47 

58 

.951771 

2.33 

.930139 

22.30 

59 

.943416  2.38 

.856494 

19.52 

59 

.951911 

2.35 

.931477 

22.33 

60 

9.943559  i  2.38 

10.8:-7665 

19.55  1 

60 

9.952052 

2.83 

10.932817 

22.40 

445 


TABLE  XXVI.— LOGARITHMIC  VERSED  SINES 


84° 

85° 

' 

Vers. 

D.  1". 

Ex.  sec. 

H 

i  ; 

Vers. 

D.  1'. 

Ex.  sec. 

D.I". 

0  9.952052 

2.33   10.932817 

22.40 

0 

9.960397 

2.30  ill.  020101 

26.40 

1 

.952192 

2.33 

.934161 

22.45 

1 

.900535 

2.28 

.021085 

20.48 

2 

.852388 

2.% 

.935508 

22.52 

2 

.960672 

2.30 

.023274 

26.57 

3 

.952473 

2.33 

.936859 

22.57 

S 

.960810 

2.30 

.024868 

26  05 

4 

.932613 

2.33 

.938213 

22.62 

4 

.960948 

2.30 

.020407 

26.73 

5 

.952753 

2.33 

.939570 

22.68 

5 

.961086 

2.28 

.028071 

26.80 

6 

.952893 

2.33 

.940931 

22.75 

6 

.961223 

2.30 

.029079 

26.90 

7 

.953033 

2.33 

.942296 

22.78 

7 

.961361 

2.28 

.031293 

26.98 

8 

.953173 

2.33 

.943663 

22.85 

8 

.961498 

2.30 

.032912 

.27.07 

9 

.953313 

2.33 

.915034 

22.92 

9 

.961636 

2.28 

.034530 

27.13 

10 

.953453 

2.33 

.946409 

22.97 

10 

.961773 

2.30 

.030164 

27.23 

11 

9.953593 

2.32 

10.947787 

23.03 

11 

9.961911 

2.28  ill.  037798 

27.33 

12 

.953732 

2.33 

.949169 

23  08 

12 

.962048 

2.30 

.039438 

27.40 

18 

.953872 

2.33 

.950554 

23.15 

13 

.962186 

2.28 

.041082 

27.50 

14 

.954012 

2.33 

.951943 

23.22 

14 

.962323 

2.28 

.042732  27.58 

15 

.954152 

2  32 

.953336 

23.27 

15 

.962460 

2.28 

.044387  27.67 

1G 

.954291 

2.  '33 

.954732 

23.33 

16 

.962597 

2.  SO 

.046047 

27.77 

17 

.954431 

2.33 

.956132 

23.38 

17 

.962735 

2.28 

.047713 

27.  H5 

18 

.954571 

2.32 

.1157535 

23.45 

18 

.962872 

2.28 

.049384 

27.93 

19 

.954710  2.33 

.958942 

23.52 

19 

.963009 

2.28 

.051060 

28.03 

20 

.954850 

2.32 

.960353 

23.57 

20 

.663146 

2.28 

.052712 

28.13 

21 

9.954989 

2.33 

10.961767 

23.  G5 

21 

9.963283 

2.28 

11.054430 

28.22 

22 

.955129 

2.32 

.963186 

23.70 

22 

.963420 

2.28 

.050123 

28.30 

23 

.955268 

2.32 

.964608 

23.77 

23 

.963557 

2.28 

.057821 

28.40 

24 

.955407 

2.33 

.966034 

23.82 

24 

.903694 

2.28 

.059525 

28.50 

25 

.955547 

2.32 

.967463 

23.90   25 

.963831 

2.28 

.061235 

28.60 

26 

.955686 

2.32 

.968897 

23.95   26 

.963968 

2.27 

.002951 

28.68 

27 

.955825 

2.32 

.970334 

24.02   27 

.904104 

2.28 

.004672 

28.78 

28 

.955964 

2.32 

.971775 

24.10  ||  28 

.964241 

2.28 

.000399 

28.88 

29 

.9561  as 

2.33 

.973221 

24.15  1  29 

.964378 

2.28 

.008132 

28.98 

30 

.950243 

2.32 

.074070 

24.22 

30 

.964515 

2.27 

.009871 

29.08 

31 

9.95G382 

2.32 

10.976123 

24.28 

31 

9.964651 

2.28 

11.071616 

29.18 

32 

.956521 

2.32 

.977580 

24.35 

32 

.964788 

2.27 

.073367 

29.28 

33 

.956660 

2.32 

.9790H 

24.42 

33 

.964924 

2.28 

.075124 

29.38 

34 

.956799 

2.30 

.980506 

24.48 

34 

.965061 

2.27 

.070387 

29.48 

35 

.956937 

2.32 

.981975 

24.55  1  35 

.965197 

2,28 

.078656 

29.58 

36 

.957076 

2.32 

.983448 

24.63  1  36 

.965334 

2.27 

.080431 

29.68 

37 

.957215 

2.32 

.984926 

24.68 

37 

.965460 

2.28 

.082212 

29.80 

38 

.957354 

2.32 

.986407 

24.77 

38 

.965607 

2.27 

.084000 

29.90 

39 

.957493 

2.30- 

.987893 

24.83  1  39 

.965743 

2.27 

.085794 

30.00 

40 

.957631 

2.32 

.989383 

24.90 

40 

.965879 

2.28 

.087594 

30.12 

41 

9.957770 

2.32 

10.990877 

24.97 

41 

9.966016 

2.27 

11.089401 

30.22 

42 

.957909 

2.30 

.992375 

25.03 

42 

.966152 

2.27 

.091214 

30.32 

43 

.958047 

2.32 

.993877 

25.12 

43 

.966288 

2.27 

.093033 

30.43 

44 

.958186 

2.30 

.995384 

25.18 

44 

.966424 

2.27 

.094859 

30.55 

45 

.958324 

2.32 

.996895 

25.27 

45 

.996560 

2.27 

.096692 

30.07 

46 

.958463 

2.30 

.998411 

25.33 

46 

.966696 

2  27 

.098532 

30.77 

47 

.958601 

2.30 

.999931 

25.40 

47 

.966832 

2.  '27 

.100378 

30.87 

48 

.958739 

2.32 

11.001455 

25.48 

48 

.966968 

2.27 

.102230 

31.00 

49 

.958878 

2.30 

.002984 

25.52 

'49 

.967104 

2'.27 

.104090 

31.12 

50 

.959016 

2.30 

.004517 

25.03 

50 

.967240 

2.27 

.105957 

31.22 

51 

9.959154 

2.30 

11.006055 

25.70 

51 

9.967376 

2.27 

11.107830 

31  .35 

52 

.959292 

2.32 

.007597 

25.78 

52 

.9(57512 

2.25 

.109711 

31.45 

53 

.959431 

2.30 

.009144 

25.  85 

53 

.907647 

2.27 

.111598 

31.58 

54 

.959569 

2.30 

.010095 

25.93  I  54 

.967783 

2.27 

.113493 

31.68 

55 

.959707 

2.30 

.012251 

26.  OJ  [  55 

.967919 

2.25 

.115394 

31.82 

56 

.959845 

2.30 

.013811 

26.10  ' 

56 

.968054 

2.27 

.117303 

31.93 

57 

.959983 

2.30 

.015377 

26.17 

57 

.968190 

2.27 

.119219 

32.07 

58 

.960121 

2.30 

.016947 

26.23 

58 

.968326 

2.25 

.121143 

32.18 

59 

.960259 

2.30 

.018521 

26.33 

59 

.968401 

2.27 

.123074  !  32.30 

60 

9.960397 

2.30 

11.020101 

26.40  ! 

60 

9.968597 

2.25 

11.125012  i  32.43 

AND  EXTERNAL  SECANTS. 


86°                      87° 

' 

Vers. 

D.  1".  Ex.  sec. 

D  1". 

:     i 
'    Vers.   D.  1".|  Ex.  sec. 

D.  r. 

0  9.96859T  1  2.25 

11.125012  32.43 

0  9.976654 

2.23 

11.257854 

42.52 

1    .908732 

2.27 

.126958 

32.55 

1 

.976788 

2.22 

.260405 

42.73 

2 

.968808 

2  25 

.128911 

32.70 

2 

.976921   2.22 

.262969 

42.95 

3 

.909003 

2.25 

.130873 

32.80 

3 

.977054  !  2.22 

.265546 

43.20 

4 

.969138 

2.27 

.132841 

32.95 

4 

.977187 

2  22 

.268138 

43.42 

5 

.969274 

2.25 

.134818 

,33.07 

5 

.977320 

2^20 

.270743 

43.67 

G 

.909409 

2.25 

.136802 

33.22 

6 

.977452 

2.22 

.273363  43.88 

7 

.969544 

2.25 

.138795 

as.  as 

7 

.977585 

2.22 

.275996  44.15 

0 

.969079 

2.25 

.140795 

33.47 

8 

.977718 

2.22 

.278645  44.38 

9 

.909814 

2.25 

.142803 

33.62 

9 

.977851 

2.22 

.281308  44.63 

10 

.909949 

2.25 

.144820 

33.73 

10 

.977984 

.2.20 

.283986  44.88 

11 

9.970084 

2.27 

11.146844 

as.  88 

11 

9.978116 

2.22 

11.286679 

45.13 

12 

.970220 

2.23 

.148877 

34.02 

12 

.978249 

2.22 

.289387  45.38 

13 

.970a54 

2.25 

.150918 

34.17 

13 

.978382 

2.20 

.292110  45.65 

14 

.97-0489 

2.25 

.152908 

34.30 

14 

.978514 

2.22 

.294849  45.92 

15 

.970024 

2.25 

.155026 

34.43- 

15 

.978617 

2.20 

.297604  46.17 

16 

.970759 

2.25 

.157092 

34.00  1  16 

.978779 

2.22 

.300374 

46  45 

17 

.970894 

2.25 

.150108 

34.73 

17 

.978912 

2.20 

.303161 

46  72 

18 

.971029 

2.25 

.161252 

34.87 

18 

.979044 

2.22 

.305964 

47.00 

19 

.971104 

2.23 

.103344 

35.03 

19 

.979177 

2.20 

.308784 

47.27 

20 

.971298 

2.25 

.105446 

35.17 

20 

.979309 

2.22 

.311620 

47.55 

21 

9.971433 

2.25 

11.107-550 

a5.as 

21 

9.979442 

2.20 

11.314473 

47.83 

22 

.971568 

2.23 

.169676 

35.48 

22 

.979574 

2.20 

.317343 

48.13 

23 

.971702  2.25 

.171805  35.63 

23 

.979706 

2.20 

.320231 

48.43 

24 

.971837  2.23 

.173943  35.78 

24 

.979838 

2.20 

.323137 

48.72 

25 

.971971   2.25 

.176090  35.93   25 

.979970 

2.22 

.326060 

49.02 

26 

.972106.;  2.23 

.178246  36.10   26 

.980103 

2.20 

.329001 

49.33 

27 

.972240  2.23 

.180412  30.27  !  27 

.980235 

2.20 

.331961 

49.63 

28 

.972374 

2.25 

.182588  ;  36.42  !  28 

.980307 

2.20 

.334939 

49.93 

29 

.972.509 

2.23 

.184773  l  36.58  !  29 

.980499 

2.20 

.337935 

50.27 

30 

.972043  2.23 

.186968  36.75 

30 

.980631 

2.20 

.310951 

50.58 

31 

9.972777  2.25 

11.189173  36.90 

31 

9.980763 

2.20 

11.343986 

50.92 

32 

.972912  2.23 

.191387  37.08 

82 

.980895 

2.18 

.347041  51.23 

33 

.97-3046  2.23 

.193612  37.25 

33 

.881026 

2.20 

.350115 

51.58 

34 

.973180  2.23 

.195847  i  37.42 

34 

.981158 

2.20 

.353210 

51.92 

35   .973314  2.23 

.198092  37.58  1  35 

.981290 

2.20 

.356325 

52.25 

30 

.97'3448  2.23 

.200347  37.77   36 

.981422 

2.20 

.359460 

52.62 

37 

.973582 

2.23 

.202613  37.93   37 

.981554 

2.18 

.362617 

52.95 

38 

.973716  2.23 

.204889  38.12   38 

.981685 

2.fiO 

.365794 

53.32 

39 

.973850  2.23 

.207170  38.28  i  39 

.981817 

2.20 

.368993 

53.68 

40 

.973984 

2.23 

.209473  38.47   40 

.981949 

2.18 

.372214 

54.07 

41 

9.974118  2.23 

11.211781  38.67   41 

9.982080 

2.20 

11.375458 

54.42 

42 

.974252 

2.23 

.214101  38.83  i 

42 

.982212 

2.18 

.  .378723 

54.80 

43 

.974J386  2.22 

.216431   39.03  :  43 

.982343 

2.20 

.382011 

55.20 

44 

.974519  2.23 

.218773  39.20  i  44 

.982475 

2.18 

.385323 

55.58 

45 

.974653 

2.23 

.221125  39.42  jl  45 

.982606 

2.18 

.388658 

55.97 

40 

.974787 

2.22 

.223490  39.58  1  46 

.982737 

2.20 

.392016 

56.38 

47 

.974920  i  2.23 

.225805  39.80  ||  47 

.982869 

2.18 

.395399 

56.80 

48 

.975054  2.23 

.228253  39.98  !i  48 

.983000 

2.18 

.398807  57.20 

49 

.975188  i  2.22 

.230(152  :  40.18  |  49 

.983131 

2.18 

.402239  57.62 

50 

.975321   2.23 

.233063  40.38  |  50 

.983202 

2.20 

.405696  58.07 

51 

9.975455  2.22 

11.235486  40.58 

51 

9.983394 

2.18 

11.409180 

58.48 

52 

.975588  2.23 

.237921  :  40.78  |  52 

.983525 

2.18 

.412689 

58.93 

53 

.975722 

2.22 

.240368  41.00   53 

.983656 

2.18 

.416225 

59.38 

54 

.975855 

2  22 

.242828 

41.20  i  54 

.983787 

2.18 

.419788  59.83 

55 

.975988 

2^23 

.245300 

41.42  1  55 

.983918 

2.18 

.423378 

60.28 

50 

.970122 

2.22 

.247785 

41.63 

50 

.984049 

2.18 

.426995 

60.77 

57 

.976255  S  2.22 

.250283 

41.  as 

57 

.984180 

2.18 

.430641 

61.25 

58 

.976:388 

2.22 

.252793 

42.07 

58 

.984311 

2.18 

.434316 

,61.  73 

59 

.976521 

2.22 

.255317 

42.28 

59 

.984442 

2.18 

.438020 

62.22 

60  9.970654  2.23   11.257854  42.52 

60  9.984573  2.17  11.441753  62  73 

447 


TABLE  XXVI.— LOGARITHMIC  VERSED  SIGNS  AND  EXTERNAL 
SECANTS. 


88 

0 

! 

89° 

t 

Vers. 

D.  r. 

Ex.  sec. 

q+l 

Vers. 

D.  1". 

Ex.  sec. 

Q+J 

15.  29* 

15  30* 

0 

9.984573 

2.17 

11.441753 

9086  : 

0 

9.992334 

2.13 

11.750498 

6801 

1 

.9847-03 

2.18 

.445517 

9215  ! 

1 

.992482 

2.15 

.757925 

6929 

2 

.984834 

2.18 

.449311 

9345 

2 

.992611 

2.13 

.765477 

7056 

3 

.984965 

2.18 

.453137 

9474  ' 

3 

.992739 

2.15 

.773158 

718-1 

4 

.985096 

2.17 

.456994 

9603 

4 

.992868 

2.13 

.780978 

7312 

5 

.985226 

2.18 

.460883 

9732  i 

5 

.992996 

2.13 

.788926 

7440 

6 

.985357 

2.17 

.464805 

9HO-2 

6 

.993124 

2.15 

.797-022 

7507 

7 

.985487 

2.18 

.468761 

9991 

7 

.993233 

2.13 

.805268 

7005 

8 

.985618 

2.17 

.472751 

4-120 

8 

.993:381 

2.13 

.8136(58 

7823 

9 

.985748 

2.18 

.476775 

0249 

9 

.993509 

2.13 

.822229 

7950 

10 

.985879 

2.17 

.480834 

0378 

10 

.993637 

2.13 

.830956 

8078 

11 

9.986009 

2.18 

11.484929 

0507 

11 

9.993765 

2.15 

11.839858 

8205 

12 

.986140 

2.17 

.489061 

0636 

12 

.993894 

2.13 

.848940 

8333 

13 

.986270 

2.17 

.493230 

0765 

13 

.994022 

2.13 

.858211 

8460 

14 

.986400 

2.18 

.497437 

0894 

14 

.994150 

2.13 

.867079 

8588 

15 

.986531 

2.17 

.501683 

1023 

15 

.994278 

2.13 

.877351 

8715 

16 

.986661 

2.17 

.505968 

1152 

16 

.994406 

2.13 

.887239 

8843 

17 

.986791 

2.17 

.510293 

1281 

17 

.994534 

2.13 

.897350 

8970 

18 

.986921 

2.17 

.514659 

1410 

18 

.994662 

2.12 

.907697 

9097 

19 

.937051 

2.17 

.519066 

1539 

19 

.994789 

2.13 

.918290 

9225 

20 

.987181 

2.17 

.523516 

1668 

20 

.994917 

2.13 

.929141 

9352 

21 

9.987311 

2.17 

11.528010 

1797 

21 

9.995045 

2.13 

11.940264 

9479 

22 

.987441 

2.17 

.532548 

1925 

22 

.995173 

2.13 

.951672 

9607 

21 

.987571 

2.17 

.537131 

2054 

23 

.995301 

2.12 

.963381 

9734 

21 

.987701 

2.17 

.541760 

2183 

24 

.995428 

2.13 

.975408 

9862 

2~) 

.987831 

2.17 

.546437 

2312 

25 

.995556 

2.12 

11.987769 

9988 

26 

.987961 

2.17 

.551161 

2440 

26 

.995683 

2.13 

12.000485 

0 

27 

.988031 

2.17 

.555935 

2369 

27 

.995811 

2.13 

.013578 

0243 

28 

.988221 

2.15 

.560759 

2698 

28 

.995939 

2.12 

.027069 

0370 

29 

.98S330 

2.17 

.565634 

2826 

29 

.996066 

2.12 

.040984 

0497 

30 

.988480 

2.17 

.570561 

2955 

30 

.996193 

2.13 

.055352 

0024 

31 

9.988610 

2.15 

11.575542 

3083 

31 

9.996321 

2  12 

12.070202 

0751 

32 

.988739 

2.17 

.580578 

3212 

32 

.996448 

2!  13 

.085569 

087'8 

33 

.988869 

2.15 

.585670 

3:340 

33 

.996576 

2.12 

.101490 

1005 

34 

.988998 

2.17 

.590819 

3469 

34 

.996703 

2.12 

.118008 

1132 

35 

.989128 

2.15 

.596027 

3597 

35 

.996830 

2.12 

.185168 

1259 

36 

.989257 

2.17 

.601295 

3726 

36 

.996957 

2.13 

.153024 

1386 

37 

.989337 

2.15 

.600625 

3854 

37 

.997085 

2.12 

.171634 

1513 

38 

.989516 

2.17 

.612018 

3983 

38 

.997212 

2.12 

.191066 

1(540 

39 

.989046 

2.15 

.617475 

4111 

39 

.997339 

2.12 

.211396 

1767 

40 

.9897r5 

2.15 

.622998 

4239 

40 

.997466 

2.12 

.232712 

1894 

41 

9.989904 

2.17 

11.628589 

4368 

41 

9.997593 

2.12 

12.255116 

2020 

42 

.990034 

2.15 

.634250 

4496 

42 

.997720 

2.12 

.278723 

2147 

43 

.990163 

2.15 

.639982 

4(524 

43 

.997847 

2.12 

.303674 

2274 

44 

.990292 

2.15 

.645788 

4752 

44 

.997974 

2.12 

.3:30129 

2401 

45 

.990421 

2.15 

.651668 

4881 

45 

.998101 

2.12 

.358285 

2527 

46 

.990.550 

2.15 

.657626 

5009 

46 

.998228 

2.12 

.388375 

2654 

47 

.990679 

2.15 

.663663 

5137 

47 

.998355 

2.10 

.420686 

2781 

48 

.990808 

2.15 

.669781 

5265 

48 

.998481 

2.12 

.455575 

2907 

49 

.990937 

2.15 

.675984 

5393 

49 

.908608 

2.12 

.493490 

3034 

50 

.9.91066 

2.15 

.682272 

5521 

50 

.9!  1ST.  '35 

2.12 

.535009 

3161 

51 

9.991195 

2.15 

11.688649 

5649 

51 

9  99RS62 

2.10 

12.5S0893 

3287 

52 

.991324 

2.15 

.695U7 

5777 

52 

.998988 

2.12 

.0321  73 

3414 

53 

.991453 

2.15 

.701679 

5905 

53 

.999115 

2.10 

.690291 

8540 

54 

.991582 

2.13 

.708338 

008:3 

54 

.999241 

2.12 

!  757364 

3067 

55 

.991710 

2.15 

.715097 

6161 

:>:> 

.999368 

2.10 

.8:36072 

3793 

56 

.991839 

2.15 

.721958 

6289 

56 

.999494 

2.12 

32.933708 

31KO 

57 

.991968 

2.13 

.728925 

6417 

57 

.999621 

2.10 

13.058774 

4046 

58 

.992096 

2.15 

.736002 

6.545 

58 

.999747 

2.12 

.234991 

4172 

59 

.992225 

2.15 

.743192 

6673 

no 

.099S74 

2.10 

.536148 

4299 

eo'* 

9.992354 

2.13 

11.750498 

6801 

63 

10.000000 

2.10 

Inf.  pos. 

4426 

15.30*. 

15.81* 

TABLE  XXVII.— NATURAL  SINES  AND  COSINES. 


0° 

1°  . 

2°    i 

3° 

40 

Sine  '  Cosin  i 

Sine  |  Cosin 

Sine  Cosin 

Sine  |  Cosin  j 

Sine  Cosin 

0 

ToOOOoTOneT  .01745 

999Sr> 

703490  ! 

.99939 

.052341.99803 

.06976 

.99756 

60 

1 

.10029  One. 

.01774 

.99984 

.03519! 

.99938 

.05263:.  99801 

.07005 

.99754  59 

.00058:  One.   .01803 

.9998-1 

.03548 

.99937 

.o:,292  .99800  .07034 

.99752  58 

3  .00087!  One.  i  .01832 

.99983' 

.03577 

.99936; 

.05321  .99858  !  .07063 

.99750  57 

4  i.  00116:  One.  j.  .01802 

.99983 

.03600 

.99935 

.05350!.  99857,  .07092 

.99748  50 

5  1  .00145  One.  '  .01891 

.99982 

.03635 

!  99934 

.05379  .99855  .07121 

.99740  55 

6 

.00175  One.   .01920 

.999S2 

.03604 

.99933 

.05408  .99854 

.07150 

.99744  54 

.00204!  One.   .01949 

.99981 

.03093 

.99932! 

.05437  .99852 

.07179 

.99742  53 

8 

.00233  'One.  |  .01978 

.99980 

.03723 

.99931 

.05466  .99851 

.07208 

.99740!  52 

9 

.00262:  One.   .02007  .99980 

.03752 

.99930 

.05495  .99849  j  .07237 

.99738!  51 

10 

.00291  One.   .02036  .99979 

;.  03781 

.99929; 

.05524  .99847 

.07266 

.99730,  50 

11 

.  00320  !.  99999  .02065  .99979 

1.03810 

.99927"  .05553 

.99846 

.07295 

.99731  49 

12 

.  00349  .  99999  .  Oil  194  .  99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.997315  48 

13 

.00378  .99999  .02123  .99977 

.03808 

.99925 

.05011 

.99842 

.07353 

.99729!  47 

14 

.  00407  .  99999  .  02152  >  .  99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727  40 

15 

.00436  .99999  .021811.99970 

.03926 

.99923 

.05069 

.99839 

.07411 

.99725  45 

16 

.0040.-,  .99999  .02211  .99976 

.08955 

.99922 

.05098 

.99838 

.07440 

.99723 

44 

17 

.00495  .99999  .022401.99975 

.03984 

.99921 

.05727 

.99836 

.07469 

.99721 

43 

18 

.00524  .  '.19999  .0220'.)  .999T4 

.04013 

.99919! 

.05756 

.99834 

.07498 

.99719 

42 

19 

.  00553  .  !  >9!  )!  IS  .  Oii'.KS  !  .  99974 

.04042 

.99918! 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00582  .99998  1.02327  j.  99973 

.04071 

.99917; 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611  .99998 

.02356!.  99972 

.04100 

.99916: 

!  .05^44 

.99829 

.07585 

.99712 

39 

22 

.(KM)  40  .99998  .02:385  1.99972 

.04129 

.99915 

.05873 

.99827! 

.07614 

.99710 

38 

23 

.  00669  .  99998  .  0241  4  j  .  99971 

.04159!.  99913  i 

!  .05902 

.99826 

.(•7043 

.99708 

37 

24 

.  00698  t.  99998  .02443  .99970 

.04188 

.99912 

;  .05931 

.99824 

.07672 

.99705 

36 

25 

.  00727  !  .  99997  .  02472  !  .  99969 

.04217 

.99911 

i  .05960 

.99822 

.07701 

.99703 

35 

26 

.00756  .99997  .0*501 

.99909 

.04246).  99913  : 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785.99997  .02530 

.99908 

!.  04275 

.99909 

.06018 

.99819 

.07759 

.99099  33 

28 

.00814  .99997  .02500  .99907  .04304  .99907 

.06047 

.99817 

.07788 

.99696  32 

29  .  00844  '.  99996  .  02589  :  .99966 

.04333 

.99906 

'  .06076 

.99815 

.07817 

.99694 

31 

30 

.00873  .99996:  .02618  .99968 

.04362 

.99905 

.06105 

.99813  .07846  .99692 

30 

31 

.  00902  !.  99996  j 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812  .07875  .99689 

29 

32 

.00931  .99990  :  .02076  .99904 

.04420 

.99902 

i  .06163 

.99810  .07904  .99687 

28 

33 

.00960  .99995  :|  .02705  1.99963 

.04449 

.99901 

.  06192  .  99808  .  07933  .  99685 

27 

34 

.00989  .99995  .02734  .99963 

.04478 

.9990(3 

.06221 

.99806  .  07962  :.  99683 

26 

35 

.01018  .99995  .02703  .99962 

.04507 

.99898 

!  .06250 

.99804  .07991!.  99680 

25 

36 

.01047  .99995  !  .02792 

.99961 

.04536 

.99897 

:.  00279  .99803 

.  08020  j.  99078 

24 

.01076  .99991 

.02821 

.99900 

1.045651.99896 

.00508 

.99801 

.08049!.  99676 

23 

38 

.01105  .99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99799 

.080781.99073 

22 

39 

.01134  .99994 

.02879 

.99959 

1.04623 

.99893 

.06366 

.99797 

.08107  !.  99071 

21 

40 

.01164  .99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795  .08136  .99068 

20 

41 

.  Oil  93  !.  99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793  .081  65  '.99666 

19 

42 

.01222  .99993 

.02967 

.99956 

.04711 

99>"H9 

.  06453  .  99792  .  08194  i  .  99664  !  18 

43 

.01251  .99992 

.02996 

.99955 

.04740 

.'91)888 

.06482 

.99790 

.08223 

.99661  17 

44 

.01280  .99992 

.03025 

.99954 

.04709 

.99886 

.06511 

.99788 

.08252 

.99659  16 

45 

.01309  .99991 

.03054 

.99953 

..04798 

.99885 

.06540 

.99786  1.08281 

.99657 

15 

46 

.01338  .99991' 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784 

f*8310 

.99654  14 

47 

.  01367  :.  99991 

.03112 

.99952 

1.04856 

.99882 

.06598 

.99782 

!08339 

.99652  13 

48 

.01396  .99990 

.03141  .99951 

.04886 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425:.  99990 

.03170  .99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647  11 

50 

.01454!.  99989 

.03199  .99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644  10 

51 

.  01483  !.  99989 

.03228 

.99948 

.04078 

.  99876  ' 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513  .099S9 

.03857 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.01542:.  99988 

!  03236 

.99946 

!.  05030 

.99873 

.06773 

.99770 

.08513 

.99037 

7 

54 

01571  .99988 

.03316 

.99945 

.05059 

.99872 

!.  06802 

.99768 

.08542 

.99035 

6 

55 

01600  .99987 

.03345 

.99944 

.05088 

.99870 

.00831 

.99706 

.08571 

.99032 

5 

56 

.01629  .99987  >• 

.03374 

.99943 

.05117 

.99809 

.06860 

.99764 

.08600 

.99630 

4 

57 

.01658  .99986 

.03403 

.99942 

.05146 

.99867 

.OOSS9  .99762 

.08629 

.99627 

3 

58 

.01087  99986 

.03432 

.99941 

.08175 

.99800 

.06918  .99760! 

.08658 

.99025 

2 

59 

.01716  .99985 

.03401 

.99940 

i.05205 

.99864 

.06947  .99758 

.08687  .99022 

1 

60 

.01745  .99985 

.03490 

.99939 

|.  05234 

.99863 

.06976  .99756 

.087161.99619 

0 

f 

Cosin  Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

~ 

89° 

88° 

87° 

86° 

85° 

449 


TABLE  XXVII.— NATURAL  SINES  AND  COSINE3. 


5° 

6° 

7° 

.  8' 

9° 

' 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

f 

0 

.08716 

.99619 

T0453 

.99452 

.12187 

.99255 

.13917 

.99027 

715643 

.98769 

60 

1 

.08745 

.99617 

.10482 

.99449 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764 

59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019;  .15701 

.98760 

58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015!  .15730 

.98755 

57 

4 

.08831 

.99609 

.10569 

.99440 

.12:302  .99240 

.14033 

.99011 

.15758 

.98751 

56 

5 

.08860 

.99607 

.10597 

.123311.99237  .14061 

.99006 

:  .15787 

.98746 

55 

6 

.08889 

.99604 

.10626 

iium 

.12360  .992*3  1  .14090 

.99002 

.15816 

.98741 

54 

7 

.08918 

99602 

.10655 

.99431 

.123891.99230  i  .14119!  .98998 

.15845 

.98737 

53 

8 

.08947 

.99599 

.10684 

.99428 

.12418  .99226 

.14148 

.98994 

.15873 

.98732 

52 

9 

.08976  .99596 

.10713 

.99424 

.12447  .99222 

.14177 

.98990 

1.15902 

.98728 

51 

10 

.09005  .99594 

.10742 

.99421 

.12476  .99219 

.  14205  j.  98986 

.15931 

.98723 

50 

11 

.09034  .99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

.15959 

.98718 

49 

12 

.01)063  .99588 

.10800 

.99415 

.12533  .99211 

.14263 

.98978 

i  .15988 

.98714 

48 

13  i  .09092  .99586 

.10829 

.99412 

.12562  .99208 

.1  4292  !.  98973 

i  .16017 

.98709 

47 

14  !  .  09121  i.  99583 

.10858 

.99403; 

.12591  .99204 

.14320 

.98969 

i.16046 

:  1)870  J 

46 

15  .09150  .99580 

.10387 

.12620  .99200 

.143491.98965 

.16074  .98700 

45 

16  .09179  .99578 

.10916 

'.in  103 

.12649!.  99197 

.14378 

.98961 

.16103 

.98(595 

44 

17  .09208  .99575 

.10945 

.93393 

.126781.99193' 

.14407 

.98957 

.  161  32  !.  98690 

43 

1  8  1.  092371.  99572 

.10973 

99  59  i 

.12106  .99189 

.14436 

.98953  .16160 

.  9SOS(5 

42 

19  i  .09266  .99570 

.11002 

!  99391 

.12735  .  99186  '• 

14464  .98948  .1(5189  .98(581 

41 

20 

.09295 

.99567 

.11031 

.9339,); 

.12764  .99182 

.14493 

.98944;  .162181.98676 

40 

21 

.09324 

.99564 

.11060 

.99^(5 

.12793  .99178 

.14522 

.98940'  .16246 

.98671 

39 

22 

.09353 

.99562 

.11089 

.993S3 

.12822 

.99175 

.145511.98936  .16275 

.98667 

38 

23 

.03:382 

.99559 

.11118 

.13851 

.99171 

.14580 

.98931  .16304 

.98662 

37 

24 

.09411  .99556 

1.11147  .99377 

.12880 

.99167 

.14608 

.981*27  .16333 

.98657' 

36 

25  .09440  .99553 

.1117(5 

.93374 

.12908 

.93163 

.14(537 

.1)8923  .16361 

.98652 

35 

26  .09469 

.99551 

.11205 

.99370 

,12937 

.99160 

.1  1000 

.98019  .16390 

.9S01S 

34 

27  .09498  .99548 

.11234 

.99  lf>7 

.129(56 

.99156 

.11695 

.98914  .16419 

.98(543 

33 

28  i  .095271.99545 

.11263  .99364! 

.12995 

.9915-2 

.14723 

.98910  .16-147 

.98638 

32 

29  .09556^.99542 

.11231 

.99330 

.13024 

.99148 

.14752 

.981100  .1(5476 

.986*3 

31 

30 

.09585 

.99540 

.11320 

.99357, 

.1:3053 

.99144 

.11781  .98902  1.16505 

.98629 

30 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141  ! 

.14810 

.98897  .16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.93351 

.13110 

.99137 

.14838 

98893   1(5502 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.148(57 

!!  98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436  .93344; 

.13163 

.99129 

.1489(5 

L  98884 

.16(520 

.9S009 

26 

35 

.09729 

.99526 

.11465 

.93341 

.13197 

.99125! 

.11925 

.98880:  .16648 

.98(504 

25 

36 

.09758 

.99523 

.11494 

.99337 

.99122 

.14954 

.98876  .16677 

.98600 

24 

37 

.09787  S.  99520 

.11523 

.93334 

!  13254 

.991181 

.14982 

.  98871  .16706 

.98595 

23 

38  1  .09816  .99517 

.11552 

.99331 

.13283 

.99114 

.15011 

!.  98867  .16734 

.98590 

22 

39  .098451.99514 

.11580 

.93327 

.13312 

.99110 

.15040 

.988631  .16763 

.98585 

21 

40 

.09874  .99511 

.11609 

.99324 

.13341 

.99106 

.15069 

:.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102 

.15097 

1.98854  .16820 

.98575 

19 

42 

.09932 

.99506 

.11667 

.99317 

.13399  .99098 

.15126 

.98849  .16849 

.98570 

18 

43 

.09961 

.99503 

!.  11696 

.99314 

.134271.99094 

.15155 

;.  98845.  .10878 

.98505 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13456  .99091 

.15184 

.98841'  .1(590(5 

.98561 

16 

45 

.10019 

.99497 

.117541-99307 

.1:3485  .99087 

.15212  .98836  .16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99:303 

1  13514 

.99083 

.15241 

.<:8S;;2  .169(54 

.98551 

14 

47 

.10077 

.99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.H8827'  .16992 

.  98.546 

13 

48 

.101061.99488 

.11840 

.99297 

.13572 

.95)075 

.15299  .98823  .17021 

.98541 

12 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.15327 

.!)8818  .17050 

.9853(5 

11 

50 

.10164 

.99482 

.  11893  i.  99230 

.13629 

.  99067 

.15356  .C8814  .17078 

.98531 

10 

51 

.10192 

.99479 

.11927 

L  99286 

.1  3658  '.  99063 

.15385 

.98809  ".  7107 

.96526 

9 

52 

.10221  .99J7< 

:  .119561.992*3  .1368T  .99059 

.15414 

.98805  .  7136 

.98521 

8 

53 

.10250  .9947': 

!  .11985 

.93279  !  .13716  .99055 

.16442 

98800  .  71(54 

.9851(5 

7 

54 

.10279  .9917( 

.12014 

.99376 

.13744  .UlH-,1 

.15471 

.9879(5  .  7193 

.98511 

6 

55 

.10308  .99467 

.12043 

.99272 

.13773  .1)1)047 

.  5500  .9N791  .  7222  .9S50I5 

5 

56 

.  10337  .  99  4(5- 

.12071 

.992(59 

.13802  .99043  i  .  5529 

.98787  .  726C 

9S501 

4 

57 

10366  .99461!  .12100 

.992(55 

.138:31  .99039 

.  5557 

.98782  .  7271 

.U8190 

3 

58 

.10395  .99458 

.12129 

.99262 

138(50  990:35  .  558C 

98778  l  .  7308 

.984D1 

2 

59 

.10424  .99455 

•  .12158 

.99258 

.138S9  .99031   .  .-(iir 

.98773  .17381) 

.98486 

1 

60 

.10453  .99452 

i.  12187 

i.99255 

.13917 

.99027  .15(513 

;.  98769  .1  7:565 

.  9S  181 

!  0 

Cosin 

Sine 

1  Cosin  |  Sine 

Cosin 

Sine 

Cosin 

i  Sine   Cosin 

Sine 

i 

84° 

83° 

82°    il  x  81°   [1    80° 

450 


TABLE  XXVII.— NATURAL  SINES  AND  COSINES. 


10° 

|    11°   1 

12° 

13° 

14° 

' 

Sine  Cosin 

Sine 

Cosin  I 

Sine  i  Cosin 

Sine  Cosin 

Sine 

Cosin 

0 

.17305  .1)848) 

.19081 

.98163 

.20791  .97815 

.22495  .97437 

.24192 

T97030 

60 

1 

.17393  .98476 

1.  19109 

.98157 

.20820  .97809 

.22523 

.97430 

.24220 

.97023 

59 

2 

.17422  .98471 

.19138 

.98152  .20848:.  97803: 

.22552 

.97424 

.24249 

.97015 

58 

3 

.17451  .98466 

.19107 

.98146 

.20877  .97797 

.22580 

.97417 

.24277  .97008 

57 

4 

.17479  .98461 

.19195 

.98140 

.20905  .97791 

.22608 

.97411  .24305  .97001!  56 

5 

.17508  .98455 

.19224 

.98185] 

.209:33  .97784 

.22637 

.97404  .24333 

.86994  55 

6 

.17537  .98450 

.19252 

.9812!) 

.20962;.  97778 

.22065 

-97398  !  .24362 

.96987  54 

7 

.17565  .98445 

.19281 

.98124 

.20990 

.97772  .22093 

.9739l!  .24390 

.96980 

53 

8 

.17591  .98440 

.19309 

.98118 

.21019 

.97766   22722 

.97384:  .24418 

.96973  52 

9 

.17623  .98435 

.  19338 

.98112 

.21047 

.97700 

.22750 

.97378  .24446 

.96966  51 

10 

.1765JL  .98430 

.19366 

.98107: 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.176801.98425  .19395 

.98101 

.21104 

.97748 

.22807  .97365 

.24503 

.96952 

49 

12 

.  17708  !.  98  420 

.19423 

.21132 

.97742 

.22835  .97358 

.24531 

.96945 

48 

13 

.17737  .98414 

.  19452  I.98C90 

.21161 

.97735 

.22363  .97351 

.24559 

.96937 

47 

14 

.17700  .9.8409 

.19481 

.98084 

.21189 

.97729 

.2289:2  .97345 

.24587 

.96930  46 

15 

.17794  .98404 

.195091.98079 

.21218 

.97723 

.22920  .97*38 

.24615 

.90923'  45 

16 

.178231.98399 

.19538 

.98078 

.21246 

.97717 

.,22948  j.  97331 

.24644 

.96916 

44 

17 

.17832  .93394 

.19500 

.98067 

.21275 

.97711 

22977 

.97325 

.24672 

.96909 

43 

18 

.17X80  .9S381) 

.19593 

.93061 

.21303 

.97705 

!  2:3005 

.97318 

.24700 

.96902  42 

19 

.17909  .983831 

.19623 

.98056 

.21331 

.97698 

.230331.97311 

.24728 

.96894!  41 

20 

.17937:.  98378  ; 

.19552 

.98050,  .21360 

.97692 

.23002  1.97304  .24756 

.  96887  !  40 

21 

.17966  .98878 

.19680 

.98044  !  .21388 

.97686  .23090  .97298  .24784 

.96880  39 

.17!)'.<5  .!)S3li8 

.19709 

.93039  .21417 

.97680 

.23118!.  97291!!  .24813 

.96873  38 

23 

18()'*3  ')SJ')'* 

.19737 

1K)33  .21445 

.97673 

.23146 

.972.-U   .24841 

.96866  37 

24 

,'lS052  \  1)8357 

.19766 

.98027  .21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

.13081  .98352 

.19794 

93021  .21502 

.976t51 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109  .98347 

.  19323 

.93316  .21530 

.97055 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18138  .98341 

.19851 

.98910 

.21559 

.97648 

.23260 

.97257 

.241)54 

.96837 

33 

28 

.18100  .98330 

.19880 

.98004 

.21587 

.97612 

.23288 

.97251 

.24982 

.96829 

32 

29 

.18195  .93331: 

.19908 

.979:)-! 

.21616 

.97633 

.23316 

.97244 

.25010 

.90822 

31 

30 

.18224;.  98325; 

.19937 

.97932 

.21644 

.97030 

.23:345 

.97237 

.25038 

.96815 

30 

31 

.18252  '.98320 

.19965 

.97937; 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.1823  1  .93315 

.19994 

.97331  .21701 

.97017 

.23401 

.97223 

.25094 

.96800 

28 

33 

.1330!)  .98310 

.20322 

.!)7i>;5  .21729 

.97611 

.23423 

.97217 

.25122 

.96793]  27 

34 

.18338  .1K301 

.20051 

.97969  .21758 

.97604 

.2:3458 

.97210 

.25151 

.967861  26 

35 

.188671.98299 

.23379 

.!)7;«3  .21786 

.97533 

.23486 

.1)7203  .25179 

.96778  25 

30 

.18393  .98294 

.23103 

.97958  .21814 

.97592 

.23514 

.97196l|  .25207 

.96771  24 

37 

.18124  .98238 

.20136 

.97952  .21843 

.97535 

.23542 

.97189  |  .25235 

.96764  23 

38 

.18452  .98283 

.20165 

.97946  |  .21871 

.97579 

.23571 

.97182  l  .25263 

.96756!  22 

39 

.18481  i.98277 

.20193 

.97940  .21839 

.97573 

.23599 

.'37170  .25291 

.  96749  j  21 

40 

.18509  .98272 

.20222 

.97934  .21923 

.97566 

.23627 

.97109  .25320 

.96742  20 

41 

.1  8538  .  98267 

.20250 

.971)23  .21956 

.97560 

.23656 

.97162  .25348 

.96734!  19 

42 

.18507  .{W261 

.2037'!) 

.<)r:)22  .21985 

.97553 

.23084  .97155  .25376 

.96727!  18 

43 

.18595  .98250 

^20307 

.97916  .22013 

.97547; 

.23712 

.97148  ;  .25404  .96719  17 

44 

.18024  .1)8250 

.20330 

.97910  .22041 

.97541 

.23740 

.97141  ii  .25432  1.90712  16 

45 

.181552  .9821-) 

.20364 

.97903 

.22070 

.97534 

.23709 

.97134'!  .25460  .967051  15 

46  .18081  .1IS240 

.  20393  L  97899  ! 

.22093  .97528 

.23797 

.97127  II  .25488 

.96697!  14 

47 

.18710  .98234 

.20121 

.97833 

.221261.97521 

.2:3825 

.9712(7, 

.25516!.  966901  13 

48 

.18738  .93229 

.204,50 

.97887; 

.22155  .97515 

.23853 

.97113 

.25545  .96082  12 

49 

.187'07  .98223 

.20478 

.97331 

.22183  .97508  .23882  .1)7100 

.25573  .96673  11 

50 

.18795  .98218 

.20507 

.97875 

.22212.97502  .23910  \  .97100  .25001'.  96667  10 

51 

.18S21  .98212 

.20535 

.97809 

.22240  .97496 

.239381.97093  .25629  .96660  9 

52 

.18852  .93207 

.97863  .22208  .97489 

.23966 

.970861 

.25057  .  90053  i  8 

53 

.18881  .1)82,11 

!  20592 

.97857  .22297  .97483 

.23995 

.1)707'!)  .25085  .  96645  1  7 

54 

.18910  .98196 

.20820 

.1)7851  .22325  .97476 

.24023 

.97072  .25713  .96638  6 

55 

.189381.98190; 

.20649 

.97845  .22353  .97470  .24051  .97065  .25741  '.90030 

5 

56 

.18907  .98185 

.20677 

.9783!)  .22382  .97463  .24079  .97058  .23709  .96623  4 

57 

.18995  .98179 

.20706 

.97833  .22410  .97457  .24108  .1)7051  .257!)8  .90015  3 

58 

.19024  .98174 

.207:34 

.97827  .22438  .97450  .24130  .97044  .25820  .90(508  2 

59 

.19052  .98168 

.20763 

.97821  .22407.97444  .24104  .97037  .25854  .96600  1 

60 

.19081  .<)S  103 

.20791 

.97815  .22495  .97437 

.2111)2  .97030  .25882  .96593  0 

Cosin  ,  Sine 

Cosin 

Sine  Cosin  Sine 

Cosin 

Sine 

Cosin  Sine 

" 

L 

79°    ! 

7tf°    !!    77° 

76° 

75° 

451 


TABLE  XXVII.— NATURAL  SINES  AND  COSINES. 


15° 

16° 

17° 

18°       19° 

' 

Sine  Cosin 

Sine  i  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

.25882 

.96593 

.27564  .96126  S  .29237 

.95630, 

~  30902 

.95106 

.325.57 

.94552 

60 

1 

.25910  '.1658.-, 

.27592  .96118 

.jJ'.tttM 

.95622 

.30929 

.<»5097 

.32584 

.94542 

59 

2  .25938.9(5578  .27620  .96110  ;  .29293 

.95613 

.30957 

.95088 

.;woi2 

.94533 

58 

3  .25966  .96570  .27648  .90102  .29321 

.95605 

.30985 

.95079 

.32639 

.94523 

57 

4  .25994  .9fi562  .27676  .96094  .29348 

.95596 

.31012 

.95070 

.32667 

.94514 

56 

5  26022  96555  .27704.960861.29376 

.95588 

.31040 

.95061 

.32694 

.94504 

55 

6  .260501.96547 

.27731  .960781  .29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7  .26079  .96540 

.27759  -96070  1;.  29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8  .26107  .96532 

.27787  .96062  .29460 

.95562 

.31123 

.95033 

.32777 

.94470 

52 

9  .26135  .96524 

.27815  .96054  .29487 

.95554 

.31151 

.95024 

.32804 

.94400 

51 

10  -2IH63  .W517 

.27843  .9604(5  .29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11  .  26191  |.  96509 

.27871  -.96037  i  .29543 

.95536 

.31206  .95006  .32859 

.94447 

I!) 

12  .26219  .96502 

.27899  .96029  i  .29571 

.95528 

.31233  .94997!  .32887 

.944:38 

48 

13  .26247  .96494 

.27927  .96021  .29599 

.95519 

.31261  .94988  .32914 

.94428 

47 

1  4  .  26275  .  96488  j  .  27955  .  960  1  3  .  29626 

.95511, 

.31289  .94979  .32942 

.94418 

46 

15  .26303.964791.27983.96005 

.29654 

.95502 

.31316  .94970 

.:-W'.«i9 

.94to:) 

15 

16  i  .  26331 

.96471  .28011  .9599? 

.29682 

.95493 

.31344  .94961  .32997 

.94399 

44 

17  .26359 

.96463  i.  28039  .95989 

.29710 

.95485 

.31372  .94JI52  .33024 

.94890 

43 

18  .26387  .96456  .28067  .95981 

.29737 

.95476; 

.31399!.  94943  .33051 

.94380 

42 

19  !.  26415  .96448 

.28095  .95972 

.29765 

.95467 

.31427  .94933  .33079 

.94370 

41 

20 

.26443  .96440 

.28123  .95964 

j  .29793 

.95459 

.  31454  j.  94924 

.33100 

.94361 

40 

21 

.26471 

.96433 

.28150  .95956 

.29821 

.95450 

!.  31482  .94915 

.33134 

.94&51 

39 

22 

.26500 

.96425 

.28178  .95948 

.29849 

.  95441  i 

.31510  .94906 

.83161 

.94342 

i  38 

23 

.26528 

.96417 

.28206  .95940 

'.29876 

.95433 

.31537  .94897 

.33189 

.94832 

1  37 

24 

.26556 

.96410, 

.28234  .95931 

i  .29904 

.95424 

.31565  .94888 

.33216 

.94822 

36 

25  .26584 

.96402 

.28262  .95923 

;  .29932 

.95415 

.31593;.  94878  .33244 

.94313 

35 

26  1.26612 

.96394 

.28290  .95915 

.29960 

.95407 

''  .31620  .94869  .33271 

1.94303 

34 

27  .26640 

.96386 

.28318  .95907 

.29987 

.95398 

i  .31648 

.94860  .33298 

1.94293  33 

28  .26668 

.96379 

.28346  .95898 

j.  30015 

.95389 

1  .31675 

.94851  .33326 

;.  94284 

32 

29  .26696 

.96371 

.28374  .95890 

.30043 

.95380 

L31  703  .94842  .33353 

.94274 

31 

30  .26724 

.96363 

.28402  .95882  ;  .30071 

.95372 

.31730.94832  .33:381 

.94264 

;  30 

31  .26752 

.96355 

.28429  ,.95874  .30098 

.95363 

.31758  .94823  .33408 

.94254 

!29 

32  .26780 

.96347; 

.28457  .95865 

.30126 

.95354 

.:!!>(•>  .94814  .33436 

.94245 

1  28 

33  I  .26808 

.96340 

.28485;.  95857 

.301K4 

.95:345 

.31813  .94805  .33463 

!  942ft 

27 

34  !  .26836 

.96.332 

.28513;.  95849 

.30182 

.95337 

.31841 

.94795  .33490 

.94223 

26 

35  j  .26864 

.9(5324 

.285411.95841 

.30209 

.95328 

I  .31868 

.94786!  .33518 

.94215 

i  25 

36  1  .26892 

.96316 

.28569i.95832 

.30237 

I.  95319 

.31896 

.94777 

.33545 

.94906 

24 

37  1  .26920 

.96308 

.28597:.  95824 

.30265 

1.95310 

.31923 

.94708 

.33573 

.94196 

23 

38  !  .26948 

.98301 

.  28625  L  9581  6 

.30292 

:.  95301 

.31951 

.94758 

.33600 

.94186)  22 

39  1  .26976 

.96293 

.28652  .95807 

.30320 

.95293 

.31970 

.94749 

i  .33627 

.94170 

21 

40  .27004 

.96285; 

.28680  .95799 

.30340 

.95284 

.32006 

.94740 

.33655  .94107 

20 

41 

.27032 

.96277 

.28708  .95791 

.30370 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.270601.96269 

.28736  .95782 

.30403 

.95266 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261; 

.28764  .95774 

.30431 

.95257 

.32089  .94712 

.33737 

.94187 

i  17 

44 

.27116 

.96253 

.28792  .95766 

i  .30459 

.95248 

!  .32116  .94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28820  .95757  ;  .30480 

.95240 

.32144  1.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

.28847;.  95749 

.30514 

.95231 

.32171 

.94684 

.3:3819 

.94108 

14 

47 

.27200 

.96230 

.288751.95740 

.30542 

.95222 

.32199 

.94674  .33846 

.94098 

13 

48 

.27228 

.96222 

.28903  .95732 

.30570 

.95213 

.32227 

.9460.5  .33874 

.94088 

i  12 

49 

.27256 

.96214 

.28931  .95724 

.30597 

.95204 

.32254 

.94656   33<>01 

.94078 

i  11 

50 

.27284 

.96206 

.28959  .95715 

.30625 

.95195 

.32282 

.  9464d  p.  33929 

.94068 

10 

51 

.27312 

.96198 

.28987  .95707 

.30653 

.  95186  : 

.32309 

.94637  .33956 

.94058  9 

52 

.27340  .96190 

.29015  .95698 

.30680 

.  95177  , 

.32337 

.94027  .33983 

.94049 

8 

53 

.27368  .061P-2 

.29042  .95690 

.30708 

.95168 

.82364.94618  .34011 

.94039  7 

54  1.27396  .96174 

.29070  .95681 

.30736 

.95159 

.32392  .94609  .34038 

.94029 

6 

55 

.27424  .9616;! 

.29098  .95673 

.30763 

.95150 

.324191.94599  .34065 

.94019 

5 

56 

.27452  .96158 

.29126:.  95664 

.30791 

.95142 

.:!•,'  117  .94590  .34093 

.94009 

4 

57 

.27480  .96150 

.29154  .95656 

.30819 

.  95133 

.82474 

.93580  .34120 

.93999 

3 

58  .27508  .96142 

.29182  .95647 

.30846 

.95124 

.32502 

.94571  .34147 

.  93989  i  2 

59  .27536  .96134 

.29209  .95639  .30874 

.95115 

.32529  .94561  .34175 

.93979 

1 

60  .27564  .96126 

.2JW37  .  95630  i  .30902 

.95106 

.325571.94552  .31202 

.93969 

0 

Cosin;  Sine  ' 

Cosin  Sine 

Cosin 

Sine" 

Cosin  Sine 

Cosin 

Sine 

/ 

74° 

73° 

72° 

71°       70° 

452 


TABLE  XXVII.-NATURAL  SINES  AND  COSINES. 


20° 

21° 

22° 

23° 

24° 

' 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

Sine 

Cosin 

~o 

.34202  .93909 

.86837 

.93858 

.37461  .92718 

.39073  .92050 

.40674 

.91355 

60 

1  .34229  .'.mm  .&5864 

.93348 

.37488  .92707! 

.39100  .920391 

.40700 

.91343 

59 

2  .34257  .513949  .35891; 

.93337' 

.37515  .  92697  i 

.39127 

.92028 

.40727 

.91331 

58 

8  '.342S4  .93939  .35918 

i  93327 

.37512  .  92686  : 

.39153 

.92016 

.40753 

.91319 

5? 

4  .  343  11  .93929  .35945 

.93316 

.37569  .92675 

.39180  .92005! 

.40780 

.91307 

56 

5  i.  34339  .93919  1  .35973 

.93:  106 

.37595  .92664 

.39207  .91994 

.40806 

.91295 

55 

6  1.34366  .98909  .36000 

.93295 

.37622  .92653 

.39234  .91982: 

.40833 

.91283 

54 

7  .34393  .93899!  .36027 

.'.KkKi 

.37649  .92642 

.39260 

.91971 

.40860 

.91272 

53 

8  .34121  .93SS9  .30054 

.93274 

.37(57(5  .92631 

.39287 

.91  959  ; 

.40886 

.91260 

52 

!)  .344  IS  .9:JS7i) 

.36081  .i)3264 

.37703!.  92620 

.39314 

.91948 

.40913 

.91248 

51 

10  .344  75  .93869 

.36108 

.93253 

.37730  .92609 

.39341 

.91936 

.40939 

.91236 

50 

11  .34503  .93859 

.36135 

.93243 

.37757  '.92598 

.39367 

.91925 

.  40966  i.  91  224  !  49 

12  .34530  .93849 

.36162 

.93232 

.37784  .925871 

.39394 

.91914 

.40992 

.91212 

48 

13  .34551'  .H:5S;J9 

.3(il90 

.1*3222 

.37't!ll  .92576! 

.39421 

.91902 

.41019 

.91200 

47 

14  .34581  .93S29 

.36217 

.93211 

!37S38  .92565 

.  39448  i.  91  891 

.41045 

.91188 

46 

15  .34012  .1)381!) 

.36241 

.93201! 

.37865  .92554 

.39474  (.91879 

.41072 

.91176 

45 

1(5  .34039  .93S09 

.36271 

.93190 

.37892  .92543 

.39501 

.91868 

.41098 

.91164 

44 

17  .34066^.93799 

.36298 

.93180 

.37919  .92532 

.39528 

.91856 

.41125 

.91152 

43 

18  !.  34691  .93789 

.36325 

.93169 

.37946  .92521 

.39555 

.91845 

.41151 

.91140 

42 

19  1.  34721  1.93775) 

i  3(5352 

.93159 

.37W3  .92510 

.39581 

.91833 

.41178 

.91128 

41 

20  1.34748  .93769 

.36379 

.93148 

.37999  .92499 

.39608  .91822 

.41204 

.91116 

40 

21  .34775  .93759 

.36406 

.93137 

.38026!.  92488 

.39635  .91810 

.41231 

.91104 

39 

22  .34803  .9374S 

.36434 

.93127 

.SK053  .92477 

.39661  .91799 

.41257 

.91092 

38 

23  .34830  .93738 

.36461 

.93116! 

.86080  .98466 

.39688  .91787 

.41284 

.91080 

37 

21 

.34857  .93728 

.36488 

.93100 

.381071.92455 

.39715 

.91775 

.41310 

.91068 

36 

25 

.34884  .'.WIS 

.36515 

.  93095  i 

.38134  .92444 

.39741 

.91764 

.41337 

.91056 

35 

26 

.31912  .937'OS  .36512 

.!i:5i  »KJ  .38161  .92432 

.39768 

.91752 

.41363 

.91044 

34 

27 

.34939  .9369S  .:-J(356!) 

.93074  .38188  .924.21 

.39795 

.91741 

.41390 

.91032 

33 

28 

.34966  .«.)36SS  .36591; 

.9:3063  .38215  .92110 

.39822 

.91729 

.41416 

.91020 

32 

29 

.34993  .931)77  .3(562-3 

.93052 

.:-.8241  .92399 

.88848 

.91718 

.41443 

.91008 

31 

30 

.35021  .93667  .36650 

.93042 

.38268  .!  12388 

.39875 

.91706 

.41469 

.90996 

30 

31 

.350  IS  .9365? 

.36677 

.93031 

.38295  1.92377 

.39902 

.91694 

.41496 

.90984 

29 

82 

.35075  .93047  .36704 

.93020  .38322  .92366 

.39928 

.916^3  .41522 

.90972 

28 

33 

.35102  .93637  .36731 

.93010 

.38349  .92855 

.39955 

.91671 

.41549 

.90960 

27 

34 

.35130  .9.J626  -3(i758 

.92999 

.38376  '.9234  3 

.39982 

.91660 

.41575 

.90948 

26 

35 

.3515;-  .93616 

.3(5785 

.92988  .38403J.  92332 

.40008 

.91648 

.41602 

.90936 

95 

36 

.35184  .93(KI6 

.36812 

.9297S  !  88480  .92821 

.40035 

.91636 

.41628 

.90924 

24 

37 

.;i-)2  11  93596 

.36839 

.92!)(i7  ..'58456  .92310 

.40062 

.91625 

i  .41655 

.90911 

23 

38 

.353391.93585 

i  3158(57 

.92956  .38488  1.92299 

.40088 

.91613 

.41681 

.90899 

22 

39 

.352;iii  .93575 

.36894 

.92945  .38510  .92287 

.40115 

.91601 

.41707 

.90887 

21 

40 

.35293  .93565 

.36921 

.92935 

.38537  .92276 

.40141 

.91590 

.41734 

.90875 

20 

1 

41 

.35320  .93555 

.36948 

.92924 

.  38564  i.C2265 

.40168 

.91578 

.41760 

.90863 

19 

42 

.35347  .93541 

.36975 

.92913 

.  38591  i.  92254 

.40195 

.91566 

.41787 

.90851 

18 

43 

.35375  .9:5534 

.37002 

.92902 

.88617  .92243 

.40221 

.91555 

.41813 

.90839 

17 

44 

.35402  .93524 

.37029 

.92892 

.38644  .92231 

.40248 

.91543 

.41840 

.90826 

16 

45 

.35429  .93514 

.37056 

.92881  ! 

.38671  .92220 

.40275 

.91531 

.41866 

.90814 

15 

46 

.35456  .93503 

.37083 

.92870 

.38698  .92209 

.40301 

.91519  |  .41892 

.90802 

14 

47 

.35484  .93493 

.37110- 

.92859 

.387251.92198 

.40328 

.91508* 

.41919 

.90790 

13 

48 

.35511  .93483 

.37137 

.92849 

.38752  .92186  j  .40355 

.91496 

.  41945  i.  90778 

12 

43 

.3553S  .93472 

.37164 

.92838 

.38778,.92175 

.403811.91484 

.41972 

.90766 

11 

50 

.35565  .93462 

.37191 

.92827 

.38805  j.  931  04 

.40408  '.91472 

.41998 

.90753 

10 

51 

.35592  .93452 

.37218 

.92816 

.38833  .92152  .40434  .91461 

.42024 

.90741 

9 

52 

.35619  .93441 

.37245 

.92805 

.38859  .92141! 

.40461  .91449 

.42051 

.90729 

8 

53  '.  35647  .93431 

.37272 

.92794 

.38886  .92130 

.40488  .91437  i  .42077 

.90717 

7 

54  .35I574  ..9342:)  .37299 

.92784  !  .38912:.  921  19  .40514.91425  .42104 

.90704 

6 

55  .35701  .93410  .37326 

.92773  .38939  .92107  .40541  .914141  .42130 

.90692 

5 

56  .3572S  .93400 

.37353 

.  92762  .889(515  .92096  .405(57  .91402  .42156 

.90680 

4 

57  :.  35755  .933-!) 

!  37380 

.92751  :  .88993  .92085  .40594  .91390  .42183 

.90668 

3 

5S 

.3578-2  .9337!»  .37407 

.92:40  .raNyo  .(fc»073  .40(521  .91378  .42209 

.90655 

2 

59 

.35S10  .933(58  .37434 

.92729 

.3<tmO  .92062:  .40647  .91366 

.42235 

.90643 

1 

60 

.35837  .93358  .37401 

.92718 

.39073  .92050  .40674  .91355  1.42262 

.90631 

0 

f 

Cosin  Sine  ,  Cosin 

Bine 

Cosin  j  Sine 

Cosin  Sine 

Cosin 

Sine 

j 

69°       68° 

67° 

66° 

65° 

453 


TABLE  XXVII.— NATURAL  SINES  AND  COSINES. 


25° 

26° 

27° 

28°    i 

29° 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  i 

Sine  i  Cosin 

/ 

0 

.43882  1.90631 

.43837 

.89879 

.45399 

.89101 

.41)1)47 

.88295 

.48481  .S74C-.' 

60 

1 

.422881.90818 

1  .43863 

.89867 

.45425 

.89087 

.46973 

.88281 

.48506  .87448 

59 

2 

.  42315  i.  9060(5 

i.  43889 

.89854 

.45451 

.89074 

.46999 

.88267; 

.48532  .87434 

58 

3 

.423411.90594 

1  .43916 

.89841 

.45477 

.89061 

.47024 

.88254' 

.48557  .87420 

57 

4 

.42367  .90582 

<  .43942 

.89828 

.45503 

.89048 

.47050 

.  88240  : 

.48583  .87406  56 

5 

.42394  .90509 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608  .87391 

55 

6  .43420  .90557 

;  .43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634  .87377 

54 

7  j.  42446  .90545 

!  .44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659  .87363 

53 

8  1.  42473  .  9053-2 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684  .87349 

52 

9 

.42  499i.  90520 

.44072 

.89764 

.45632 

.88981 

.4717M 

.88172 

.48710  .87-335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735  .87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761  1.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.8^928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47300 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089  .48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075  •!  .48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062:1.48913 

.87221 

43 

18  .42736  .90403 
19  i.  42762  !.9039t5 

.44307 
.44333 

.89649 
.89636 

.45865 
.45891 

.88862  .47409 
.  88848  H.  47434 

.88Q48!  .489:38  .87207 
.  88034  :  1.48964  1.  87193 

42 
41 

20  I  .42788 

.90383 

.44359 

.89623  .45917 

.  88835  H.  47460 

.88020  .48989 

.87178 

40 

21 

.42815 

.90371  .44385 

.  89610  ! 

.45942 

.88822 

.47486 

.88006!'  .49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597! 

.45968 

.88808!  .47511 

.87993  .49040 

.87150 

38 

23  .42367 

.90346 

.44437 

.89584  .45994 

.88795  .47537 

.87979  1  .49065 

.87136 

37 

24 

.42894 

190834 

.44464 

.  83571  1 

.4(5020 

.887-82  .47562 

.  87965  j  .49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768;!  .47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88T85 

.47614 

.87937! 

.49141 

.87093 

34 

27 

.42972 

.90296  .44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

98 

.42999 

.90284  .44568 

.89519 

.46123 

.88728 

.47665 

.87909  : 

.49192 

.8706-1 

32 

29 

.43025 

.91271 

.44594 

.89506! 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

•30 

.43051 

.90259  .44620 

.89493: 

.46175 

.88701 

,47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246  .44646 

.89480 

.46201 

.88688 

.47741 

.87868: 

.49268 

.87021 

29 

32 

.43104 

.90233  .44672 

.89467 

.46226 

.88674: 

.47767 

.87854  .49293 

.87007  28 

33 

.43130 

.90221  .44698 

.89454 

.46252 

.88661 

.47793 

.87840  .49318 

.86993 

27 

34 

.43156 

.90233^ 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826  .49344 

.86978 

28 

35 

.43182 

.90196 

.44750 

.89428 

.46304 

.8863-4 

.47844 

.87812  .49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798! 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.478951.87784:  .49419 

.86935 

23 

38 

.43261  .90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770  .49445 

.86921 

22 

39 

.43287  .90146 

.44854 

.89376 

.40407  1.88580 

.47946 

.87756!  .49470  .86906 

21 

40 

.43313 

.90133; 

.44880 

.89363 

.46433  .88566 

.47971 

.87743  .49495 

.86892 

20 

41 

.43340 

.90129 

.44906 

.89350 

.464581.88553 

.47997 

.87729  .49521 

.86878 

19 

42 

.433061.90103 

.44932 

.89337 

.  46484  !.  88539 

.48022 

.87715:  .495461.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.  46510  i.  88526 

.48048 

.87701  .49571  .86849 

17 

44 

.43418 

.90082 

.44984! 

.89311 

.46536:.  88512 

.48073 

.87(587  .49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.48099 

.87673!  .49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87(559'  .49647 

.86805 

14 

47 

.43497 

.90045  .450621 

.89272 

.46613 

.88472 

.48150 

.87645  .49672 

.86791 

13 

48 

.43523 

.90032  .450881 

.89259 

.46639  .88458 

.48175 

.87(531  .49697 

.86777 

12 

49 

.43549 

.90019  .45114! 

.89245 

.46664i.88445 

.48201 

.87(517  .49723 

.86762 

.11 

50 

.43575 

!  90007 

.  45140  | 

.89232 

.46690 

.88431 

.48226 

.87003  .49748 

.86748 

10 

51 

.43602 

.89994 

.45166! 

.89219 

.46716 

.88417  .48252 

.87589  '  .49773 

.86733 

9 

53 

.43628 

.89931  .45192 

.89203;  .46742 

.  88404  i  .48277 

.87575J  ,.49798 

.86719 

8 

53 

.43654 

.89938  .45218 

.89193  .46767 

.8839;)  .48303 

.87561,  .49^24 

.86704 

7 

54 

.43680 

.89956 

.45243 

.  89180  [i  .46793 

.88377  .48328 

.87540  .49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167:  .46819 

.883(53  .48354 

.87532, 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153  .46844 

.88349  .48379 

.87518 

!  49899 

.86661 

4 

57 

.43759  .89918 

.45321*.  89  140  .46870 

.88336  .4,8405 

.87504  .49924 

.86646 

3 

58 

.43785;.  89905 

.45347 

.89127  .46896 

.88822 

.48430 

.87490  .49950 

.86632 

2 

59 

.  4381  ll.  89892 

.453731 

.89114  .46921 

.88308 

.48456 

.87  4  70  .49975 

.86617 

1 

60 

.43837  .89879 

.45399! 

.89101  .-41)947 

.88295  .4SIS1 

.87-1(52  .50000  .86603 

0 

/ 

Cosin  |  Sine 

Cosin  |  Sine 

Cosin  |  Sine 

Cosiu 

Sine   Cosin  |  Sine 

/ 

~ei°~~ 

63° 

62° 

61 

60° 

454 


TABLE  XXVII.— NATURAL  SINES  AND  COSINES. 


30°   !    31°    1    32°   i 

33° 

34° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine  'Cosin 

Sine  I  Cosin 

Sine  Cosiii 

/ 

~o 

.50000 

.86603 

.51504 

.85717 

.52992  '.84805' 

.54464 

.83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017 

.84789 

.54488 

.83851 

.55943 

.82887 

59 

2 

.50050 

.86573 

.51554 

.85687 

.530-11 

.  84774  j 

.54513 

.83835 

.55968 

.82871 

58 

3 

.50076 

.86559 

.51579 

.85672 

.530C6 

.84759 

.54537 

.83819 

.55992  .82855 

57 

4 

.50101 

.86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.82822 

55 

6 

.50151 

.86515 

.51653 

.85627 

.531401.84712 

.54610 

.83772 

.56064 

.82806  54 

7 

.50176 

.86501 

.51078  .85612 

.53164  1.84697; 

.54635 

.83756 

.56088 

.82790  53 

8 

.50201 

.86486 

.51703 

.85597 

.53189  .84681' 

.54659 

.83740J 

.56112 

.82773  52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84666 

.54683 

.83724 

.56136 

.82757  51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.837u8 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708  48 

13 

.50327 

.86413 

.51828 

.85521 

.  53312  j.  84604 

.54781 

.83660 

.56232 

.82692  47 

14 

.50352 

.86398 

.51852 

.85506 

.533371.84588: 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.533611.84573 

.54829 

.83629 

.56280 

.82659  45 

1C 

.50403 

.86369 

.51902 

.85476 

.  53386  i.  84557 

.54854 

.83613 

.56305 

.82643  44 

17 

.50428 

.86354 

.51927 

.85461 

.534111.84542 

.54878 

.83597 

.56329 

.82626  43 

18 

.50453 

.86:340 

.51952 

.8544(3 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610  42 

19 

.50478 

.86325 

.51977 

.85431 

.53460 

.  84511  i 

.54927 

.83565 

.56377 

.82593!  41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495  .54951 

.83549 

.56401 

.82577  40 

21 

.50528 

.86295 

.52026 

.85401 

.58609 

.84480 

.54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

.86281 

.52051 

.ST>;K> 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544i  38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448: 

.55024 

.83501 

.56473 

.82528!  37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.  84433  i 

.550481.83485 

.56497 

.82511  36 

25 

.50628 

.HI)-,':',?' 

.52126 

.85340 

.53607 

.84417: 

.55072 

.83469 

.56521 

.82495  35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.56545 

.  82478  i  34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437[ 

.56569 

.82462  33 

28 

.50704 

.86192 

.52200 

.85294 

.536811.84370 

.55145 

.834211 

.56593 

.82446!  32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405! 

.56617 

.82429  31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339: 

.55194 

.83389 

.56641 

.82413  30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84303 

.55242 

.83356 

.56689 

.82380  28 

33 

.50829 

.86119 

.52324 

.85218 

.5:3804 

.84292 

.55266  .83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277; 

.55291 

.833241 

.56736 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261;. 

.55315 

.83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.  55339  i.  83292 

.56784 

.82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55363 

.83276 

.56808 

.82297  23 

38 

.509541.86045 

.52448 

.85142 

.539261.84214 

.55388 

.&3260 

.56832 

.82281  |  22 

39 

.50979 

.860:30 

.52473 

.85127 

.539511.84198 

.55412 

.83244 

.56856 

.82264 

21 

40 

.51004 

.86015 

.52498 

.85112 

.53975  .84182; 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

'.52522 

.85096 

.54000  .84167 

.55460 

.83212  .56904 

.82231 

•19 

42 

.51054  .85985 

.52547 

.85081 

.54024 

.84151 

.554841.831951 

.56928 

.822141  18 

43 

.51079  .85970 

.52572 

.85086 

.54049 

.84135 

.55509 

.83179 

.56952 

.82198 

17 

44 

.511041.85956 

.52597 

.85051 

.54073!.  84120 

.5b533!.83163 

.56976 

.82181 

16 

45* 

.51  129:.  85941 

.52621 

.85035 

.540971.84104 

.55557 

.83147 

.57000 

.82165 

15 

46 

.51154  1.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131 

.57024 

.82148 

14 

47 

.51179  .85911 

.52671 

.85005 

.54146 

.84072 

.556051.83115 

'.57047 

.82132 

13 

48 

.512041.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098: 

.57071 

.82115 

12 

49 

.  51229  .  85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082! 

.57095 

.82098 

11 

50 

.51254i.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

.83066; 

.57119 

.82082 

10 

51 

.51279  .85851 

.52770 

.84943 

.54244 

.84009! 

.55702 

.83050 

.57143 

.82065 

9 

52 

.513041.85836 

.52794 

.84928 

.54269  .83994 

.55726 

.83034  .57167 

.82048 

8 

53 

.513291.85821 

.52819 

.84913 

.54293J.83978 

.55750 

.83017; 

.57191 

.82032 

7 

54 

.51354  |.85806 

.52844 

.84897 

.54317  .83902 

.55775 

.83001! 

.57215 

.82015 

6 

55 

.513791.85792 

.52869 

.84882 

.54342  .83946 

.55799 

.  82985  j 

.57238 

.81999 

5 

56 

57 

.514041.85777 
.51429  .85762 

.52893 
.52918 

.84866 
.84851 

.54366  .83930! 
.543911.83915 

.55823 

.5:5847 

.82969! 
.82953 

.57262 
.57286 

.81982 
.81965 

4 
3 

58 

.51454  .85747 

.52943 

.84836 

.54415  .83899  .55871 

.82936 

.57310 

.81949 

2 

59 

.51479  .85732 

.52967 

.84820 

.544401.83883  1.55895 

.82920; 

.57334 

.81932 

i 

60 

.51504  .85717 

.52992  .84805  .  54464  |  ,  83867 

.55919 

.82904! 

.57358 

.81915 

6 

Cosin  Sine 
/ 

Gosin|  Sine 

Cosiu  j  Sine  i  Cosin 

Sine 

Cosin 

Sine 

/ 

59° 

58° 

57°    1    56° 

55° 

_J 

455 


TABLE  XXVII.— NATURAL  SINES  AND  COSINES. 


35° 

36°    | 

37° 

38°    1    39° 

Sine  !  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

.57358  .81915!  .58779 

.80902! 

,001  82  .79864"! 

.61566 

.78801 

.02932 

.77715 

60 

1 

.57381  .818!)!);  .58802 

.80885 

.60205  .79846 

.61589 

.7-8783 

.62955 

.7701)6 

59 

2 

.57405  .81882!  .58826 

.80867: 

.60228  .79829 

.61612 

.7-87(55 

.62977 

.77678 

58 

3 

.57429  .81865!-  .58849 

.80850; 

.60251  .79811 

.61635 

.78747 

.63000 

.77660 

57 

4 

.57453I.81848' 

.58873 

.80833 

.60274  .79793 

.61658|.78729 

.63022 

.77641 

56 

5 

.57477J.81  832; 

.58896 

.80816 

!  60298  !  !  79776  ! 

.61681 

.78711 

.63045 

.77623 

55 

6 

.575011.81815! 

.58920 

.80799 

.60321;.  79758; 

.61704 

.78694 

.68008 

.77605 

54' 

7 

.57524  1.81  798  : 

.58943 

.80782 

.60344!.  79741; 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548S.81782! 

.58967 

.80765 

.60367  .79723, 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622! 

.63158 

.77531 

50 

11 

.57619 

.81731 

.590371.  80713  ' 

.60437 

.79671 

.61818 

.78604' 

.63180 

.77513 

40 

12 

.57(543 

.  81714  |j.  59061  1.80696! 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698  1  .59084 

.80679 

.60483 

.79635' 

.01864 

.78568 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80662! 

.60506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.80(544 

.60529  .79600 

.61909 

.78532 

.03271 

.77439 

45 

16 

.57738  .81647 

.59154 

.80627 

.605531.79583 

.61932 

.7-8514 

.63293 

.77421 

44 

17 

.57762  .81631 

.59178 

.80610 

.605761.79565 

.61955 

.78496 

.6:3316 

.77402 

43 

18 
19 

.57786  .81614 
.57810!.  815971 

.59201 
.59225 

.80593 
.80576 

.60599  |.79547 
.60622  .79r>::0 

.61978 
.62001 

.78478 
.78460 

.68838 
.68861 

.77-384 
.77366 

41 

20 

.57833L81580I 

.59248 

.80558 

.60645  .79512 

.02024 

.78442 

.63383 

.77347 

40 

21 

.57857  .81563 

.592721.80541 

.«0608  .79494  .6204(5 

.78424! 

.63406 

.77329 

39 

22 

23 

.57881J.  81  546 
.  57-904  j.  81530  : 

.59295 
.59318 

.80524 
.80507 

.60691  .79477  !  .62069 
.607141.79459!  .62092 

.78405 

.78387' 

.634128 
.63451 

.77310 
.77292 

38 
37 

24 

.579281.815131 

.59342 

.80489 

.607381.79441!  .62115 

.78369 

.63473 

.77273 

36 

25 

.57952!.  81496! 

.59365 

.80472 

.60761 

.  79424  .62138 

.78351! 

.63496 

.77255 

35 

26 

.679761.81479 

.59:389 

.80455 

.60784 

.79406  ;  .62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.  81462  ; 

.59412 

.80438 

.60807 

.79388  .62183 

.78315 

.6:3540 

.77218 

33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.(52206 

.78297; 

.63563 

.77199 

32 

29 

.58047J.81428! 

.59459 

.80403 

60853 

.62229 

.78279- 

.63585 

.77181 

31 

30 

.58070  .81412; 

.59482 

.80-386 

.60876 

;  79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395  .59506 

.80368 

.60899 

.79318  .62274 

.78243 

.63630 

.77144 

29 

32 

.58118  .8137'S 

.5952'.) 

.80351 

.60923 

.79300  .(5-J2S17 

.78225 

.68658 

.77125 

28 

33 

.58141  .81361  .59552 

.80334 

.60945 

.7-1:282  .62320 

.78206 

.63675 

.77107 

27 

34 

.581651.81344!  .595761.80316 

.60968 

.792641  .62342 

.78188 

.63698 

.77088 

26 

35 

.581  89:.  81  327 

.59599  .80299 

.60991 

.792471  1.62365  1.7  81  70 

.63720 

.77070 

25 

36 

.58212  .81310  .59622 

8(  )-.'82 

.61015 

.79229  .62388 

.781  52  : 

.68748 

.77051 

24 

37 

.  58236  ;  .  81  293  .  59646  |  .  80264 

.010:18  .71)211  .02411 

.78134 

.63765 

.77033 

23 

38 

.58260  .81276  .59669 

.80247 

.61061  .79193  .024:!:; 

.78116 

.63787 

.77014 

22 

39 

.58283  .81259  .59693 

.80230] 

.610841.79176 

.62456 

.78098 

.03810 

.70996 

21 

40 

.  58307  j.  81242|  .59716 

.80212 

.611071.79158 

.62479 

.78079; 

.'63832 

.70977 

20 

41 

.58330  .81225 

.59739 

.801951  .611301.79140 

.62502 

.78061 

.63854 

.76959 

1!) 

42 

.58354:.  81308  .5!)7'(>:! 

.80178!  .61153  .79122 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378  .81191 

.59786  .80160 

.61176  .79105  .02517 

.78025 

.63899 

.76921 

17 

44 

.58401  .811741 

.59809 

.80143 

.61199  .79C87  .62570 

.78007 

.63922 

.70903 

16 

45 

.58425  .81157! 

.59832 

.80125 

.61222  .79069 

.62592 

.77988 

.63944  .70884 

15 

46 

.58449  .81140- 

.59856!.  80108! 

.61245!.  79051 

.62615 

.77970 

.63906 

.76800 

14 

47' 

.  58472;.  811  23  i 

.59879 

.80091! 

.61268!.  70033  .0.-JO.S8 

.77952 

.63989 

.70847 

13 

48 

.58496 

.81106i 

.59902 

.80073 

.61291  1.79016  .02000 

.  77934 

.64011 

.76828 

12 

49 

.58519 

.81089!  .59926 

.80056  .61314  .7'89!!S  .02083 

.77916 

.64033 

.76810 

11 

50 

.68545 

.81072|  .59949 

.80038;  .61337  .78980  .62706 

.77897 

.  04056 

.76791 

10 

51 

.58567 

.81055! 

.59972 

,  80021  !  .613GO  .7F962  .62728 

.77879' 

.64078 

.76772 

9 

52 

.88590 

.81038! 

.59995 

800031  .6138:3  .7S944  .<W75l 

.77861 

.01100 

.7-0754 

8 

63 

.58614 

.81021  .60019 

.799S6  .01401)  .7X5)26'  .62774 

.77843 

.64123 

.76735 

7 

54 

.58637 

.  81004  !  .60042 

.7W68  .0142!)  .7S'.H)8  .6->7!H5 

.  77824 

.64145 

!  76717 

0 

55 

.58(561 

.80987J 

.600651.79951  .61451  .7*S91  .62819 

.77806 

.64167 

.76698 

5 

56 

.58684 

.80970 

.(50089 

.79934  .61474  .78873  .0:284  2 

.77788 

.64190 

.76679 

4 

57 

.58708 

.80953  .60112 

.79916  I  .61497  .78855  .0:2804 

.777011 

.04212 

.70661 

3 

58 

.58731 

.80936 

.60135 

.79899  .61520 

.78837  .62887 

.77751 

.64234 

.76642 

2 

59 

.58755 

.80919 

.60158 

.79881  .61543 

.78819  .62!)09 

.77733 

.64256 

.70623 

1 

60 

.58779 

.80902  :  .60182  !  .79864  .615(56 

.78801 

;  .62932 

.77715 

.64279 

.76604 

0 

f 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

f 

54° 

53° 

52°       51°   i 

50° 

450 


TABLE  XXVII.—  NATURAL  SIXES  AND  COSINES. 


40°    1    41° 

42° 

43°   ' 

44° 

' 

Sine 

!  Cosin 

Sine  !  Cosin   Sine  i  Cosin 

Sine  Cosin 

Sine  Cosin 

/ 

~o 

g  l£~Q 

.76604  .65606 

.754711  .66913 

.74314 

768200  773135 

.09466  .71934  60 

i 

61301 

.76586 

.65628 

.75152 

.06935 

.74295 

.68221 

.73116 

.69487L  71914  59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508  .71894  58 

3 

.64346 

.70548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529  .71873 

57 

4 

.643681.76530  .63M4 

.75395 

.66999  .74237 

.68285 

.73056 

.C9549i.71853  56 

5 

.61390 

.76511 

.65716 

.75375 

.67021|.74217 

.68306 

.73036 

.69570 

.71833  55 

6 

.01112 

.76492  .65738 

.75356 

.67043  .74198 

.68327 

.73016 

.69591  .71813  54 

.61435 

.70473  .65759 

.75337 

.67004!.  74178 

.68349 

.72996 

.69012  .71792  53 

8 

.64457 

.76455  .65781 

.75318 

.67086  .74159 

.68370 

.72976 

.69033  .71772  52 

9  .61479 

.7043(5  .65803 

.75299 

.671071.74139 

.68391 

.72957 

.69654  .71  752  51 

10  .64501  j.76417 

.65825 

.75280 

.67129  .74120 

.68412 

.72937 

.69675  .71732  50 

11 

.64524 

.76398 

.65847 

.75231 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76380 

.65869 

.75211 

.671731.74080 

.68455 

.72897 

.6971  7  '.71  691 

48 

13 

.61568 

.70361 

.65891 

.75222 

.67194:.  74061 

.68476  .72877 

.69737U  71671  47 

14 

.64590 

70H2  .65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.697581.71650  46 

15 

.61612 

.76323  .65935 

.75184' 

.67237 

.74022 

.68518 

.72837 

.  69779  i.  71630  45 

16 

.64635 

.76301  .65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800  .71610  44 

17 

.64657 

.70-283  .65378 

.75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590  43 

18 

.64679 

.  7(5.2:57 

.68330 

.75128  .67301 

.73963 

.68582 

.72777 

.69842 

.71569!  42 

19 

.61701 

.76248 

.63022 

75107  .67323 

.73944 

.68603 

.72757 

.  69862  :.  71  549  |  41 

20 

.64723 

.76229 

.66044 

75088  |.  67344 

.73924 

.68624 

.72737 

.69883  .71529 

40 

21  .64746 

.  76210  '' 

.65088 

75039  .67366 

.73904 

.68645 

.72717 

.69904  .71508 

39 

22  .64763 

.76192 

.63388  .75350;  .67387 

.73833 

.68666 

.72697 

.  69925  !.  71488!  38 

23  .61790 

.76173; 

.63103 

.75033  .67409 

.7'3865 

.63688 

.72677 

.69946k  71468  37 

24  .64812 

.761541 

.63131 

.75911 

.67430 

.73846 

.68709 

.7'2G57 

.69966 

.71447 

36 

25  .61S3t 

.76135 

.66153 

.74932 

.67452 

.73828 

.68730 

.72637 

.69987 

.71427 

35 

26  .04S50 

.76116; 

.63175 

.74973s 

.67473 

.73803 

.68751 

.72017 

.70008 

.71407 

34 

27  .64878 

.76397 

.66197 

.74953 

.67495 

.73787 

.68772 

.72597 

.70029  .71386 

33 

28  .64901 

,76078 

.63218 

.74934 

.67516 

.73767 

.68793 

.72577 

.700491.71366 

32 

29  .64923 

.  76359  ; 

.63240 

.74915' 

.67538 

.7'3747 

.63814 

.72557 

.70070 

.71345 

31 

30  .64945 

.76041 

.66262 

.  74896  i 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

.61967 

.76022 

.63284 

.  74876  :  .67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32  .6  49  Si) 

.76003 

.63338 

.74857 

.67633 

.73033 

.68878 

.72497 

70132 

.71284 

28 

33  .65011 

.75984! 

.63327 

.74833 

.67023 

.73669 

.68899 

.724771 

.70153 

.71264 

27 

34  .05013 

!  75985 

.63319 

.74818 

.67615 

.73649 

.68920 

.  72457  ' 

.7017-4 

.7124& 

26 

35  .05,155 

.75946 

.63371 

.74793  .67636 

.73023 

.63941 

.72437; 

.70195 

.71223 

35 

36 

.65077 

.75927 

.63333 

.74783  .67688 

.73610 

.68962 

.72417 

.7C215 

.71203 

24 

37  .65100 

.75908 

.015111 

.74763 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182  23 

38  .05122 

.75889 

.684381.74741! 

.67730 

73570 

.69004 

.72377 

.70257 

.711621  22 

39 

.65144 

.75870 

.63458  .  74722  :  ,67752 

.73551 

.69025 

.72357 

.70277 

.7114l|  21 

40  .65166 

.75851 

.68480  .74703  j&"7773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41  .65188 

.75832 

.68501 

.74683:  .67795  .7X511 

.69067 

.72317 

.70319 

.71100 

19 

42  .65210 

.75813 

.63523 

.7i!i:H  .678161.73491 

.69088 

.72297 

.703391.71080  18 

43  .65232 

.75791 

.68545 

.74644! 

.67837  .73472 

.69109 

.72277 

.70360J.71059  17 

44  .65251 

.75775 

.6-3563 

.74625; 

.67859  .73452 

.69130 

.72257 

.70381  .71039  16 

45  .65276 

.75756 

.68588 

.74603: 

.67880 

.73433 

.69151 

.72236 

.70401 

.71019  15 

4(i  .65293 

.  75738 

.68610 

.74533 

.67901 

.73413 

.69172 

.72216 

.70422  .7'0998  14 

47  .63330 
48  .65342 

.75719 
.75700 

.68832 
.68(553 

.745671 

.74548' 

.  67923  i.  7.3393 
.679441.73373 

.69193 
.69214 

.72196 
.721761 

.704431.70978  13 
.70463  .70957  12 

4!»  .05301 

.75680 

.666751.74528  .67965  .73353 

.69235 

.  72156  j 

.70484i.  70937 

11 

50  .65336 

.75661 

.66697 

.74509  .67987 

.73333 

.69256 

.72136 

.70505  .70916 

10 

51  .65108 

.75642 

.68718  .74489' 

.68008 

.7asi4 

.69277 

.72116 

.70525 

.70R96 

9 

52  .65130 

.75623 

.66740  .74470 

.68029 

.73291 

.69298  .720951 

.70546  .70875!  8 

.->:!  .1)5152 

.75604 

.66762  .74451  .68051  .73274 

.  69319  i.  72075i 

.70567.70355  7 

54  i  .65474 

.75585 

.66783  .74431  .  68072  !.  73254 

.693401.72055 

.705871.70834  6 

55  .65131) 

.75566 

.Or>Si)5  .74112  .6.8093  .7*234 

.69301  .720a5 

.70608  .70813  5 

56  .65518 

.75547 

.60827  .74392  .68115  .73215 

.69382 

.72015 

.70628  .70793 

4 

57  .05510 

.75528  .60S  18  .74373  .68136  .73195 

.694031.71995 

.70649  '.70772 

3 

58  .05562 

.75509  .66870  .74353  .68157 

.73175 

.694241.71^4 

.70670  .70752 

2 

59  .65584 

.75190  .66891  .74331  .68179 

.73155 

.69445  .71954 

.70690  .70731 

1 

00  .6561)6 

.75471 

.  66913  .  74314  ',  .  68200  .  73135 

.09100 

.71934 

.  7071  1!.  70711 

0 

Cjsiu  Sine 

Cosin  Sine  Cosin 

Sine 

Cosin 

Sine  |j  Cosin  Sine 

/ 

.      49°   „ 

48°   ii    47°   1!    46°    i   45° 

457 


TABLE  XHVIIL— NATURAL  TANGENTS  AND  COTANGENTS. 


0° 

1°                         2°                         3°             i 

Tang 

Cotang 

Tang 

Cotang      Tang 

Cotang      Tang     Cotang 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811    CO 

I 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

|   .05270 

18.9755    59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1064 

.05299 

18.8711    58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7078  '57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656    56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645    55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645    54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655    53 

8 

.00233 

429.718 

.01978 

50.5485    !    .03725 

26.8450 

.05474 

18.2677    52 

g 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708    51 

10 

.00201 

343.774 

.02036 

49.1039 

.03783' 

26.4316 

.05533 

18.0750 

50 

11     .00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863    48 

18 

.00378 

2G4.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934    47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

!    .05649 

17.7015  1  46 

15  1    .0043(5 

229.182 

.02182 

45.8294 

i   .03929     25.4517 

.05678 

17.6106  i45 

16;   .00465 

214.858 

.02211 

45.2261 

j   .03958  !  25.2644 

.05708 

17.5205    44 

17     .00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

i   .05737 

17.4314  !43 

18     .00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19     .00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

|   .040?'5 

21.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700  ! 

.02357 

42.4335 

1    .04104 

24.3675 

.05854 

17.0837 

09 

22     .00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990  :38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150  !37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319    36 

25     .00727 

137.507 

.02473 

40.4a58 

.04220 

23.01)45 

.05970 

16.7496    35 

26 

.00756 

132.219 

,02502 

39.9655 

.04250 

23.5321 

.05999     16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

88 

98 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.'06058 

16.5075  132 

29 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.00087 

16.4283    31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.00  175 

16.1952    28 

:;:! 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190  i27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

86 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

3l> 

.01047 

95.4895 

.02793 

35.8006 

.04541 

£2.0317 

.06291 

15.8945 

24 

87 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

•2-} 

as 

.01105-' 

90.4633 

.02851 

85.0695 

.04599 

21.7426 

.06350 

15.7483 

•2-1 

89 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762    21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048  ;20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340  !19 

& 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

IS 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20  9460 

.06525 

15.3254    16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571  !15 

M 

.C1338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893    14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222    13 

IS 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557    12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.08671 

14.9898 

11 

50     .01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51  1  .01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52!   .01513 

66.1055 

.03259 

30.6833 

05907 

19.9702 

.06759 

14.7954 

8 

53!   .01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

rv 

54!   .01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55  1   .01600 

62.4992 

.03346 

29.8823  i 

.05095 

19.6273 

.06847 

14.6059 

5 

56!   .01629 

61.3829 

.03376 

29.6245  ! 

.05124 

19.5156 

.06876 

14.5438 

4 

57;   .01658 

60.3058 

1   .03405 

29.3711  ! 

.05153 

19.4051 

.06905 

14.4823 

8 

58!   .01687 

59.2659 

.03434 

29.1220  i 

.05182 

19.2959 

.06934 

14.4212 

g 

59     .01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60!   .01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993      14.3007 

0 

t  |  Cotangj    Tang 

Cotang 

Tang      Cotang 

Tang 

j  Cotang  i    Tang 

/ 

!           89° 

88°                       87° 

86° 

458 


TABLE  XXVIII.-NATURAL  TANGENTS  AND  COTANGENTS. 


4°             i             5°            !             6°            '!            7° 

Tang     Cotang 

Tang   i  Cotang 

Tang     Cotang      Tang  i  Cotang 

0     .06993 

14.3007 

.08749 

11.4301 

.10510     9.51480       .12278 

8.14435 

60 

1'    .07022 

14.2411 

.08778 

11.3919 

.10540  |  9.48781 

.12308 

8.12481 

59 

2     .07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

8     .07080 

14.1285 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4     .07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5     .07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

56 

6     -07108 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7|   .07197 

13.8940 

.08954 

11.1681 

.10716 

9.&3155 

.12485 

8.00948 

58 

8:   .07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

n 

.07344 

13,6174 

.09101 

10.9882 

.10863 

9.20516 

.12033 

7.91582 

48 

18 

.07373 

13.56:34 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47' 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15     .07431 

13.4566 

.09189 

10.8829  ! 

.10952 

9.13093 

.12722 

7.86064 

45 

10 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.01)277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07'548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

J2840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

£2 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

.07666 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

21 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462  |l    .11246 

8.89185 

.13017 

7.68208 

36 

2<S 

.07753 

12.8981 

;    .09511 

10.5136  j     .11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.C9541 

10.4813    1   .11305 

8.84551 

.13076 

7.64732 

38 

38 

.07812 

12.8014 

.09570 

10.4491  ||    .11335 

8.82252 

.13106 

7.63005 

32 

$9 

.07841 

12.7536 

.09600 

10.4172  i     .11264 

8.79964 

.13136 

7.61287 

81 

80 

.07870 

12.7'062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

80 

81 

.07899 

12.6591 

!    .09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

•>!) 

3S 

.07929 

12.6124 

.09688 

10.3224       .11452 

8.73172 

.18224 

7.56176 

2S 

88 

.07958 

12.5660 

.09717 

10.2913  1     .11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602  i     .11511 

8.C8701 

.13284 

7.52806 

26 

86 

.08017 

12.4742 

.09776 

10.2294       .11541 

8.66482 

.13313 

7.51132 

-.'5 

3<5 

.08046 

12.4288 

.09805 

10.1988    :   .11570 

8.64275 

.13343 

7.49465 

24 

87 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

30 

.08134 

12.2946 

.09893 

10.1080    |   .11659 

8.57718 

.13432 

7.44509 

21 

40 

,08163 

12.2505  ij   .09923 

10.0780 

;   .11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067  !J   .09952 

10.0483 

.11718 

8.53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

48 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040     9.96007 

.11806 

8.47007 

.13580 

.36389 

16 

45 

.08309 

12.0346 

.10069  i  9.93101 

.11836 

8.44896 

.13609 

.34786 

15 

46 

.08339 

11.0023 

.10099     9.90211 

.11865 

8.42795 

.13639 

.33190 

14 

47 

.08368  j  11.9504 

.10128     9.87338 

.11895 

8.40705 

.13069 

.31600 

18 

46 

.08397 

11.9087 

!   .10158     9.84482 

.11924 

8.38625 

.13698 

.30018 

12 

49 

.08427 

11.8673  |     .10187 

9.81641 

.11954 

8.36555 

.13728 

.28442 

11 

50 

.08456 

11.8262 

.10216 

9.7'8817 

.11983 

8.34496 

.137'58 

.20873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

.25310 

!) 

52     .08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

.23754 

s 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

.22204 

7 

54 

.OS573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

.20661 

6 

55 

.08602 

11.6248 

I   .10363 

9.64935 

.12131 

8.24345 

.13906 

.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205  i 

.12160 

8.22344 

.13935 

.17594 

4 

57    .08661 

11.5461 

.10422 

9.59490  l 

.12190 

8.20a52 

.13965 

.16071 

3 

58!   .08690 

11.5072 

.10452 

9.56791  i 

.12219 

8.18370 

.13995 

.14553 

2 

59 

.08720 

11.4685 

|   .10481 

9.54106  i 

.12249 

8.16398 

.14024 

.13042 

1 

60 

.08749      11.4301 

.10510 

9.51436  | 

.12278 

8.14435 

.14054 

.11537 

0 

/ 

Cotang     Tang 

Cotang 

Tang    i  Cotang     Tang 

Cotang 

Tang 

f 

85°                       84°           i 

83° 

82° 

459 


TABLE  XXVIII.-NATURAL  TANGENTS  AND  COTANGENTS. 


8°            ||            9°            |            10°                        11° 

Tang 

Cotang   |  Tang     Cotang 

Tang 

Cotang      Tang 

Cotang 

0 

.14054      7.11537       .15838 

6.31376 

.17633 

5.07128 

.19138      5.14455 

60 

1 

.14084 

7.10038 

.15S08 

6.30189 

.17603 

5.00165 

.19468      5.13058 

59 

2 

.1*113 

7.08546 

.15898 

6.29007 

.17093 

5.05205 

.19498      5.]2S(W 

58 

• 

.14143 

7.07059 

.15928 

6.27'829 

.17723 

5.64248 

.19529      5.12009 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

a 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19019 

5.09704 

54 

7 

.14202 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19049 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19080 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

.10137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

IS 

.14410 

6.93952 

f.  16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037  i48 

18 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.042ii7 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18038 

5.53927 

.19801 

5.03499 

40 

LI 

.14499 

6.89088 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734    45 

If] 

.14529 

6.88278 

.16316 

6.12899  i 

.18113 

5.52090 

.19921 

5.01971   !44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210    43 

IS 

.14588 

6.85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451    42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356" 

.20012 

4.99695  :41 

80 

.14048 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

fcl 

.14078 

6.81312 

.16465 

6.07340 

.18203 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46048 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.90090 

37 

24 

.14707 

6.77199 

.16555 

6.04051  i 

.18353 

5.44857 

.20104 

4.95945 

30 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43900 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94400 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

2H 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646  ; 

.18504 

5.40429 

.20315 

4.92349 

31 

30 

.  14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.1S564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18024 

5.36936 

.20436 

4.89330 

27 

34 

.15034 

6.63831 

.16854 

5.93335 

.  18654 

5.36070 

.20466 

4.88605 

26 

85 

.15094 

6.62523 

.16884 

5.92283 

.18084 

5.35206 

.20497 

4.878S2    25 

30 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87102  j24 

37 

.15153 

6.59321 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.80444  (23 

:s 

.15183 

6.53627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727  I  28 

31) 

.15213 

6,57339 

.17034 

5.83114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20048 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20079 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52254 

.17123 

5.84001 

.38925 

5.28393 

.20739 

4.82175 

17 

41 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18980 

5.20715 

.20800 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80008 

14 

47 

.15451 

6.47203 

.17243 

5.79344 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4!  78673 

12 

48 

.15511 

6.44720    I   .17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484  I 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77280 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.70595 

9 

58 

.  15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.  75  WO 

8 

53 

.156:30 

6.39804 

.17423 

5.73900 

.19227 

5.20107 

.21043 

4.75219 

7 

51 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

5(5 

.15719 

6.36165 

1   .17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037  ! 

.19347 

5.16863 

.21104 

4.72490 

3 

58 

.15779 

6.33701 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

59 

.15809 

6.32500 

.17603 

5.68094  ! 

.19408 

5.15250 

.21225 

4.71137 

1 

60 

.15838 

0.31375  ||   .J7G33 

5.67128  i 

.10488 

5.14465 

.21256 

4.70463 

0 

/ 

Cotang 

Tang 

Cotang  |    Tang 

Cotang  1    Tang 

Cotang 

Tang 

/ 

81°            i           80°            i           79°            i           78° 

460 


TABLE  XXVIII. -NATURAL  TANGENTS  AND  COTANGENTS. 


1 

2° 

!               1 

30 

1 

40 

1 

5° 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~0 

.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

80 

1 

.21286 

4.69791 

.23117 

4.32573 

.2491  J4 

4.00582 

.26826 

3.72771 

56 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

56 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

66 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

65 

(i 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97.627 

.27013 

3.70188 

98 

8 

.21499 

4.05138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

K 

g 

.215^9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

H) 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

•I!) 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

-IS 

18 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.272432 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

19 

.21743 

4.59927 

.2357'8 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

U 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27320 

3.65957 

43 

18 

.21804 

4.5S641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

43 

19 

.21&34 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

90 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

!W 

22 

.219:25 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

88 

2:5 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

87 

34 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

80 

25 

.22017 

4.54196 

.23834 

4.19215 

.25707 

3.89004 

.  27576 

3.62636 

85 

86 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

27607 

3.62224 

81 

87 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

!  27638 

3.61814 

83 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

2!) 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

Ml 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3  60588 

au 

81 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764' 

3.60181 

20 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

83 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

84 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

20 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

86 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

21 

87 

.22383 

4.46764 

.24223 

4.12825 

26079 

3.83449 

.27952 

3.57758 

23 

:w 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57.357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

•:i 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

1!) 

!2 

.22536 

4.437'35 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

is 

43 

.22567 

4.43134 

.24408 

4.09899 

.26266 

3.80726 

.28140 

3.55364 

17 

It 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

H; 

43 

.22G28 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

i:. 

!!i 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

M 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28566 

8.33785 

18 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

19 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

5-2 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

S 

5:5 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

S.  75388 

.28517 

3.50666 

6 

56 

.22984 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

8 

58 

.2:3026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

80 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

7 

7° 

'  7 

8? 

!           7 

5° 

\           7 

4° 

461 


TABLE  XXVIIL— NATURAL  TANGENTS  AND  COTANGENTS. 


16°                      17° 

18°           ji           19°           | 

Tang     Cotang 

Tang 

Cotang 

Tang   !  Cotang 

Tang     Cotang  | 

0 

.28675     3.48741 

.30573 

3.27085 

.32492      3.07768 

.31433      2.90421    60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524  1  3.07464 

.34465  j  2.90147  |59 

8 

.28738 

3.47977 

.30637 

3.26406 

.32556  i  3.07160 

.34498      2.89873  J58 

8 

.28769 

3.47596 

.30669 

3.26067 

.32588  !  3.06857 

.34530 

2.89600 

57 

4 

.28800     3.47216 

.30700 

3.25729 

.32621  !  3.06554 

.34563 

2.89327 

5(5 

5 

.28832     3.46837 

.30732 

3.25392 

.32653     3.06252 

.34596 

2.89055 

55 

6 

.28864     3.46458 

.30764 

3.25055 

.32685  !  3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717  |  3.05649 

.34661 

2.88511 

88 

8i  .28927 

3.45703 

.30828 

3.24383 

.32749  i  3.05349 

.34693 

2.8824C 

52 

9!   .28958 

3.45327 

.30860 

3.24049 

.32782  i  3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

4!) 

12 

.'29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13    .29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

141   .29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

40 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

48 

1(5 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

4* 

ir 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

IS 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020  1  2.85555 

42 

1!) 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052  1  2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.88169 

3.01489 

.35118 

2.84758 

99 

2* 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

M 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33206 

3.00611 

.35216 

2.83965 

86 

25 

.29463 

3.3^406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702  135 

86 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439    34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176  133 

88 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914  i  32 

2!) 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653  131 

SO 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

81 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

83 

.29685 

3.36875 

.31594 

3.16517 

.33524 

2.98292 

-.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877  ; 

.33589 

2.97717 

.35543 

2.81350 

2(5 

85 

.29780 

3.35800 

.31690 

3.15558  ! 

.33621 

2.97430 

.35576 

2.81091 

2G 

3(5 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

87 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

88 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

"2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437    !    .35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155       .35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.9487'2 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400  ! 

.33945 

2.94591 

.35904 

2.78523 

16 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47     .30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.85989 

2.78014 

13 

48     .30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49!    .30224 

3.30868 

.32139 

3.11153  ! 

.34075 

2:98468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10843 

.34108 

2.93189 

.36068 

2.77'254 

10 

SI 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

B 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

s 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354  ] 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

86 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

66 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

8 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

GO 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

/ 

Cotang 

Tang 

Cotang 

Tang    j  Cotang 

Tang 

Cotang  |    Tang 

/ 

73° 

72°                       71°           II           70° 

462 


TABLE  XXVIIL— NATURAL  TANGENTS  AND   COTANGENTS. 


!           20°           !           21° 

22°           I!           23° 

|  Tang  1  Cotang      Tang     Cotang 

Tang 

Cotang  !  !   Tang 

Cotang 

0,    .30397  i  2.74748 

!   .38386 

2.005U9 

.40403 

2.47509    1    .42447 

2.35585 

00 

1 

.36430 

2.74499 

.38-120 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

'.} 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

4 

.30530 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

6 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476    |   .42619 

2.34036 

55 

6 

.36505 

2.732G3 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

51 

7 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36601 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

5:2 

9 

.30001 

2.7:2526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.3J3881 

51 

10  |   .36737 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33093 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

1-2 

.86703 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

48 

ta 

.36836 

2.71518 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

j  i 

.36850 

2.71305 

.38854 

2.57371 

.40877 

2.44036 

.42929 

2.32943 

46 

15 

.38802 

8.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

10 

.36025 

2.70819 

.38921 

2.50928 

.40945 

2.44230 

.42998 

2.32570 

41 

17 

.36958 

g.  70577 

.38955 

2.50707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36901 

2.7033r> 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

18 

.37024 

2.70094 

.39022 

2.50266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.09853 

;   .39055 

2.50046 

.41081 

2.4:3422 

.43136 

2.31826 

40 

21 

.37090 

2.09612 

1   .39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

80 

88 

.37123 

2.09371 

i   .39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

88 

•23 

.37157 

2.09131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

31 

.37190 

2.68892 

!   .39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

31  i 

85 

.37223 

S!  68653 

i    .39233 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

20 

.37256 

2.08414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

84 

37 

.37-28!) 

2.68175 

1    .39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

S3 

2* 

.87823 

2  (57!  137 

!    .39334 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

82 

89 

.37355 

2.67700 

i    .39357 

2.510H2 

.41387 

2.41620 

.43447 

2.30167 

31 

90 

.37388 

2.07462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

80 

31 

.37422 

2.07225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

33 

.  37455 

2.C.09S'.) 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

2S 

33 

87488 

2.60752 

.30492 

2.53217 

.41521 

2.40827 

.43585 

2.29437 

27 

34     .37531 

2.66516 

!    .39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

!   .39559 

2.52780 

.41592 

2.40432 

.43654 

2.29073 

85 

31; 

.37588 

2.66046 

I    .39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.05811 

i    .39626 

2.52357 

.41600 

2.40038 

.43724 

2.28710 

•33 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

82 

39 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39045 

.43793 

2.28348  121 

40 

.37730 

2.65109' 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

90 

41 

.37754 

2.64875 

.39701 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

in 

42 

.37787 

2.64042 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

18 

.37820 

2.04410 

.39829 

2.51070 

.41805 

2.38803 

.43932 

2.27626 

IV 

44 

.37853 

2.64177 

.39802 

2.50804 

41899 

2.38668 

.43966 

2.27447 

Hi 

45  j   .37887 

2.63945 

.39896 

2.50052 

.41933 

2.38473 

.44001 

2.27267    15 

46  j    .379.20 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44030 

2.27088 

11 

47     .37058 

2.63483 

.39903 

2.50229 

.42002 

2.38084 

.44071 

2.20909 

18 

48 

.37980 

2.0*253 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.20730 

12 

1!) 

.38020 

2.03021 

.40031 

2.49807 

.4207'0 

2.37697 

.44140 

2.26552 

11 

M 

.38053 

2.62791 

.40005      2.49597 

.42105 

2.87504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49336 

.42139 

2.37311 

.44210 

2.20196 

9 

52!    .38130 

2.633358 

.40132 

2.49177  1!    .42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48907 

.42207 

2.30925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55     .3S330 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56     .38253 

2.61418 

.40267 

2  48340 

.42310 

2.30349 

.44384 

2.25309 

4 

57     .38280 

2.61190 

.40301 

2.48132 

.42345 

2.30158 

.44418 

2.25132 

3 

58     .3S3-20 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

3 

.V.)     .38353 

2.60736 

i    .40369 

2.47716 

.42113 

2.35776 

.44488 

2  24780 

1 

ui)    .:-5S3si5     2.  0.c.o!) 

.40403  j  2.47509 

.1-2!!;' 

2,85585 

.  i  1523 

2.24604 

0 

Cotang  |    Tang 

Cotang      Tang 

Cotang      Tang      Cotang      Tang 

f 

69° 

!           68° 

67°           li           66° 

463 


TABLE  XXV1IL— NATURAL  TANGENTS  AND  COTANGENTS. 


2 

4° 

1           2 

5°            ! 

2 

6° 

2 

r° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  ! 

Tang 

Cotang 

0 

.44523 

2.24(304 

'    .46631 

2.14451 

.48773 

2.05030  i 

.50953 

1.96261 

60! 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

.96120 

59 

8 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

.95979 

58 

a 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

.95838 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

.95698 

56 

5 

.44697 

2.  "23727 

.46808 

2.13639 

.48953 

2.04276  j 

.51136 

.95557 

55 

6 

.44732 

2.23553 

i   .46843 

2.1:3477 

.48989 

2.04125 

.51173 

.95417 

54 

7 

.44767 

2.23378 

|   .46879 

2.13316 

.49026 

2.0397-5 

.51209 

.95277 

53 

,s 

.44802 

2.23204 

i   .46914 

2.13154 

.49062 

2.03825 

.51246 

.95137 

52 

9 

.44a37 

2.23030 

t  .46950 

2.12993 

.49098 

2.03675 

.51283 

.94997 

51 

10 

.44872 

2.22857 

{   .4(3985 

2.12832 

.49134 

2.03526 

.51319 

.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

.94579 

48 

13 

.44977 

2.22337 

1   .47092 

2.12350 

.49242 

2.03078 

.51430 

.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51.503 

.94162 

45 

18 

.45082 

2.21819 

.47199 

2.11871 

.49a51 

2.02631 

.51540 

.94023 

44 

J7 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

.93885 

43 

IS 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

.93746 

42 

1!) 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187- 

.51651 

.93608 

41 

90 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

.93470 

40 

81 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

.93332 

39 

-•2 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

.93195 

38 

28 

.45327 

2.20619 

'.47448 

2.10758 

.49604 

2.01596 

.51798 

.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

.92920 

36 

2:, 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

.92645 

34 

27' 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

.92508 

33 

2S 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

81 

.45608 

2.19261 

.47733 

2  09498 

.49894 

2.00423 

.52094 

1.91962 

29 

;w 

.45643 

2.19092 

.47769 

2.  09J341 

.49931 

2.00277 

.52131 

1.91826 

28 

83 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

:  521  68 

1.91690 

27 

84 

.45713 

2.187'55 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

86 

.45748 

2.18587 

.47876 

2.08872 

|   .50040 

1.99841 

.52242 

1.91418 

25 

86 

.45784 

2.18419 

.47912 

2.08716 

.5007'6 

1.99695 

.52279 

1.91282 

24 

87 

.45819 

2.18251 

.47948 

2.08560 

.50113 

.99550 

.52316 

1.91147 

23 

86 

.45854 

2.18084 

.47984 

2.03405 

:    .50149 

.99406 

.52353 

1.91012 

22 

89 

.45889 

2.17916 

.48019 

2.08250 

.50185 

.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

.98972 

.52464 

.9060? 

19 

48 

.45995 

2.17416 

.48127 

2.07785 

.50295 

.98828 

.52501 

.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

.98684 

.52538 

.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

|    .50368 

.9&540 

.52575 

.90203 

16 

45 

.46101 

2.16917 

.482:34 

2.07321 

i    .50404 

1.98396 

.52613 

.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

!    .50441 

1.98253 

.52650 

.89935 

li 

47 

.46171 

2.165S5 

.48306 

2.07014 

j    .50477 

1.98110 

.52687 

.8C801 

18 

4S 

.46206 

2.16420 

.48342 

2.06860 

i    .50514 

1.97966 

.52724 

.89667 

12 

4!) 

.46242 

2.16255 

.48378 

2.06706 

i    .50550 

1.97823 

.52761 

.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

;   .50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

C.  15925 

.48450 

2.06400 

1   .50623 

1.97538 

.52836 

1.89266 

9 

52 

.413348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46:383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

51 

.46418 

2.15432 

.48557 

2.05942 

:    .50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

i    .50769 

1.96969 

.52985 

1.88734 

5 

56 

46489 

2.15104 

.48629 

2.05637 

i    .50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

:   .50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2  14777 

.48701 

2.05333 

'   .50879 

1.96544 

.53096 

1.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05183 

;    .50916 

1.96402 

.53134 

1.88205 

1 

GO 

.46631 

2.14451 

.4877'3 

2.05030 

i    .50953 

1.96261 

.53171 

1.88073 

0 

Cotang 

Taiig 

Cotang 

Tang 

s  Cotang 

Tang 

Cotang 

Tang 

/ 

€ 

5° 

i           6 

4° 

6 

,3° 

i           6 

2° 

464 


TABLE  XXVIII.— NATURAL  TANGENTS  AND  COTANGENTS. 


28°           II           29°           |l           30°                       31° 

Tang-     Corang      Tang   1  Cotang      Tang   ;€otangi|   Tang   |  Cotang 

0     .53171 

1.88073        .55431 

1.804(15        .57735      1.73205        .60086 

1.66428    60 

1     .53208 

1.87941   1     .55469 

1.802H1        .57774 

.73089 

.60126 

1.66318    59 

2     .53-246 

1.87809    ;    .55507      1.80158  i     .57813 

.72973       .60165 

1.6G209    58 

31   .53283 

1.87677    i   .55545      1.80034  1!    .57851 

.72857  i     .60205 

1.66099    57 

4     .53320 

1.87.54(5       .55583 

1.79911        .57890 

.72741 

.60245 

1.65990    56 

5 

.53358 

1.87415  ||   .55621 

1.79788  :     .57929 

.72625 

.60284 

1.65881 

55 

6 

.53395 

1.87283 

.55659 

1.79665       .57968 

.72509       .60324 

1.65772 

54 

7 

.53432 

1.87152 

.55697 

1.79542  i     .58007 

.72393  ;     .60364 

1.65663  153 

8 

.53470 

1.87021 

.55736 

1.79419  I     .58046 

.72278       .60403 

1.65554    52 

B 

.53507 

1.86891 

.55774 

1.79296 

.58085 

.72163       .60443 

1.65445  151 

10 

.53545 

1.86760    |   .55812 

1.79174 

.58124 

.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

!   .58162 

.71932 

1    .60522 

1.65228 

49 

12     .53620 

1.86499 

.55888 

!  78929 

.58201 

.71817    !    .60562  i  1.65120 

IS 

13     .53(557 

1.86369  i     .55<»2G 

.78807    i   .58240 

.71702 

:    .60602 

1.65011 

47 

14     .53094 

1.86239  '     .55964 

.78685  i1    .58279 

.71588       .60642 

1.64903 

46 

15     .53732 

1.86109       .56003 

.78563  j;   .58:318 

.71473       .60681 

1.64795 

45 

16     .537(59 

1.85979       .56041 

.78441    :    .58857 

.71358  i     .60721 

1.64687 

44 

17     .53807 

1.85850       .56079 

.78319  j     .58396 

.71244  i     .60761 

1.64579 

43 

18     .53844 

1.85720 

.56117 

.78198  I!   .58435 

.71129 

.60801 

1.64471 

42 

1'.)     .5.-JSS2 

1.85591 

.56156        .78077  |!   .58474 

.71015       .60841 

1.64363 

41 

20     .53920 

1.85462 

.56194        .77955  |j   .58513 

.70901       .60881 

1.64256 

40 

21     .53957 

1.85383       .56232        .77834  i     .58552 

.707'87 

.60921 

1.64148 

39 

22     .53995      1.85204 

.56270 

.77713       .58591 

.70673 

.60960 

1.64041 

38 

23     .54032      1.85075 

.56309 

.77592 

.58631 

.70560 

.61000 

1.63934  137 

24     .5107(1      1.84946 

.66847 

.77471 

i   .58670 

.70446 

i    .61040 

1.63826  i36 

25     .54107      1.84818 

.56385 

.77351 

.58709 

.70332 

.61080 

1.63719 

35 

26 

.54145      1.K10S9 

.56424 

.772:30 

.587'48 

.70219 

.61120 

1.63612 

34 

87 

.54183  !  1.84561 

.56462 

.77110 

.58787 

.70106 

.61160 

1.63505 

33 

88 

.5422.! 

1.84433 

.56501 

.76990 

.58826 

.69992 

.61200 

1.63398 

32 

29 

.54258 

1.84805 

.56539 

.76869 

.58865 

.69879 

.61240 

1.63292 

31 

80 

.54296 

1,84177 

.56577 

.76749 

!   .58905 

.69766       .61280 

1.63185 

30 

81 

.54333 

1.84049 

!    .56616 

.76629 

.58944 

.69653    !    .61320 

1.63079 

29 

& 

.54371 

1.8392-2    '    .56654 

.76510 

.58983 

.69541 

.61360 

1.62972 

28 

83 

.54409 

1.83794    !    .56693         .7V>390 

.59022 

.69428 

.61400 

1.62866 

27 

84 

.54446 

1.83667       .56731 

.76271 

.59061 

.69316 

i    .61440 

1.62760 

26 

85 

.54484 

1.83540    !    .507(5!) 

.76151 

l   .59101 

.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413    :    .56808  i     .76032 

.59140 

.69091 

i    .61520 

1.62548 

24 

37     .515(5(1 

1.83286 

.50816  :      .75913 

.59179 

.68979  |l   .61561 

1.62442 

23 

:>o    .5i5!)r 

1.83159    1    .56885  i      .75794 

.59218 

.68866 

.61601 

1.62336 

22 

!5'.i     .5<;;:!5 

1.83033       .56923        .75075 

.59258 

.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906       .56962        .75556       .59297 

.68643 

.61681 

1.62125 

20 

•11     .54711 

1.82780  j!   .57000       .75437    1   .50336 

.68531 

.61721 

1.62019 

19 

42     .54748 

1.82654    !   .57039.       .75319    ;   .59376 

.68419       .61761 

1.61914    18 

43     .5J7NI5 

1.82528    !   .57078        .75200 

I   .59415 

.88308    !   .61801 

1.61808    47 

44  i    .54K21 

1.82402       .57116        .75082 

.59454 

.68196 

.61842 

1.617'03 

16 

45     .51802      1.82276        .57155 

.74964 

.59494 

.68085 

.61882 

1.61598 

15 

ir,     .54JHX)      l.s-21.50        .57193        .74846 

.59533 

.67974 

.61922 

1.61493 

14 

47     .54938 

1.82025        .57232 

.74728 

.59573 

.67863  |     .614:62 

1.61388 

13 

48     .51975 

1.81899       .57271        .74610 

.59612 

.67752  l|   .62003 

1.61283 

12 

49 

.55013 

1.81774   '    .57309        .74-192 

.59651 

.67641 

.62043 

1.61179 

11 

50     .55051 

1.81649       .57318        .74375 

|   .59691 

.67530 

.62083      1.61074 

10 

51      .550.-'!) 

1.81524       .57386        .74257 

,    .59730 

.67419 

.62124      1.60970 

9 

52  i    .55127 

1.813119        .57425 

.74140 

.59770  :     .67309 

.62164      1.60865 

8 

53     .55]  (',5 

1.81274       .57464        .74022 

'   .59809 

.67198 

.62204      1.60761 

7 

51     .55203 

1.81150 

.57503 

.73905    i   .55)849       .67088 

.62245 

1.60657 

6 

55     .  5.Y;!  II 

1.81025 

.57541 

.73788 

;   .59888        .66978 

.62285      1.60553 

5 

56     .5527-9 

1.80901 

.57-580        .73671 

.59928 

.66867 

.62325      1.60449 

4- 

57     .55317 

1.80777    1   .57619  !     .73555 

••   .59967 

.66757 

.62366      1.60345 

3 

58  i   .55335 

1.80653    1   .57657  j     .73438       .60007 

.66647 

.62406 

1.60241 

2 

59  i    .55393 

1.80529 

.57696        .7*321        .60046      1.  665138 

.62446  i  1.60137 

1 

GO      55431 

1.80405  i|   .57735        .73205  j     .60086      1.66428 

.62487     1.60033 

0 

t 

Cotang 

Tang 

Cotang  ,    Tang     ;  Cotang  j    Tang 

Cotang     Tang 

61° 

60°                       59°           !!           58° 

465 


TABLE  XXVIII. -NATURAL  TANGENTS  AND  COTANGENTS. 


32°                         33°                         34°                         35° 

Tang     Cotang      Tang     Cotang   | 

Tang     Cotang       Tang 

Cotang 

0 

.02487      l.r,00:;:; 

.(141)11 

1.53980 

.07451 

1.48250 

.70021 

1.42815 

00 

1 

.025-27      1.51)930 

.64983 

1.53888 

.67493 

1.48103 

.70004 

1.42726 

59 

2 

•68568 

1.59820 

.05024 

1.53791 

.67530 

1  48070 

.70107 

1.42038 

58 

3 

.02608 

1.59723 

.05005 

1.53693 

.67678 

1.47977 

.70161 

1.42550    ;>7 

4 

.02049 

1.59020 

.<;.->  100 

1,58695 

.67620 

1:47885 

.70194 

1.42462    HO 

5 

.G2G89      1.59517 

.05148 

1.53497 

.07003      1.4771)2 

.70288 

1.42374    55 

6 

.02730      1.59114 

.65189 

1.53400 

.07705      1.470W 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.05231 

1.53302 

.07748 

1.47007 

.70325 

1.42198 

63 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70388 

1.42110 

52 

9 

.02852 

1.59105 

.05314 

1.53107 

.07832      1.47422 

.70412 

1.43022 

51 

10 

.62892 

1.59002 

.G5&55 

1.53010 

.07875 

1.47330 

.70455 

1.41934 

50 

11 

.02933 

1.53900 

.05397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.02973 

1.58797 

.05438 

1.52810 

.67960 

1.47140 

.70.542 

1.41759 

48 

13 

.03014 

1.58695 

.05480 

1.52719 

.68002 

1.47053 

.70586 

1.  4H72    47 

14!    .03055 

1.58593 

.05521 

1.52022 

.68045 

1.40902 

.70029 

1.41.-.81     40 

15  1   .O30.)5 

1.58490 

.05503      1.52525 

.08088 

1  146870 

.70073 

1.4MD7    45 

IB!    .03130 

1.58388 

.05004      1.52429 

.68130 

1.407'78 

.70717 

1.41  409 

44 

17     .03177 

1  .58280 

.05041) 

1.52332 

.68173 

1.40080 

.70760 

1.41322 

43 

18     .63217 

1.58184 

.G5G88      1.52235 

.08215 

1.  46595  > 

.70804 

1.41235  [42 

19     .63-258 

1.58083 

.05729 

1.52139 

.08258 

1.40503 

.70848      1.41148  !41 

20 

.03299 

1.57981 

.65771 

1.52043 

.08301 

1.40411 

.70891 

1.41001  140 

21 

.63340 

1.57879 

.65813 

1.51940  ' 

.08343 

1.40320 

.70935 

1.40971    39 

22 

.63380 

1.57778 

.65854 

1.51850 

.08:380 

1.46229 

.70979 

U40887    38 

23 

.03421 

1.57'G76 

\    .05890 

1.51754 

.08429 

1.40137 

.71023 

1.40SOO    37 

24 

.03402 

1.57575 

.051)38 

1.51058  ! 

.68471 

1.40040 

.71066 

1.40714 

30 

25 

.63503 

1.57474 

.05980 

1.51502  ] 

.08514 

1.45955 

.71110 

1.40627 

85 

26 

.63544 

1.57372 

\    .00021 

1.51400  ; 

.08557 

1.40864 

.71154 

1.40540 

'!4 

27 

.63584 

1.57271 

.06063 

1.51370 

.08600 

1.45773 

.71198 

1.40454 

33 

28 

.03025 

1.57170 

.66105 

1.51275 

.08042 

K45682 

.71242 

1.40867 

32 

29 

.63606 

1.57009 

.66147 

1.51179 

.08085 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.50909 

.66189 

1.51084 

.08728 

1.45501 

.71329 

1.40195    80 

31 

.63748 

'1.50808 

.66230 

1.50988 

.08771 

1.45410 

.71373 

1.40109 

29 

32 

.03789 

1.50707 

.60272 

1.50893 

.08814 

1.45320 

.71417 

1.40022    28 

33 

.03830 

1.50007 

.00314 

1.50797 

.08857 

1.45229 

.71401 

1.3!)!«(;     27 

34 

.63871 

1.56566 

.00350 

1,50702 

.08900 

1.45139 

.71505 

1.39850    2(i 

35 

.63912 

1.56400 

.00398 

1.50007 

.08942 

1.45049 

.71549 

1.  397(51    2.-> 

30 

.G31>:>3 

1.50300 

.06440 

1.50512 

.08985 

1.44958 

.71593 

1.39679    21 

37 

.03994 

1.50265 

.00482 

1.50417 

.09028 

1.44808 

.71637 

1.39593    23 

38 

.04035 

1.56165 

.00524 

1.50322 

.09071 

1.44778 

.71681 

1.39507     2.2 

39 

.64076 

1.56065  1     .00500 

1.50228 

.09114 

1.44688 

.71725 

1.39421     21 

40 

.64117 

1.55966    1   .66008 

1.50133 

.69157 

1.44598 

.71769 

1.39330  |20 

41 

.64158 

1.55806  !     .66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250    19 

42 

.64199 

1.55700         OOOH2 

1.49944 

.69243 

1.44418 

.71857 

1.39105     IS 

43T    .04240 

1.55606 

.00734      1.49849 

.69:286 

1.44329  i     .71901 

1.31!0;i)     17 

44 

.04281 

1.55567 

.00776 

1.49755 

.69329 

1.44239       .71946 

1.38994    10 

45 

.04322 

1.  55467 

.66818 

1.490.01 

.09372 

1.44149 

.71990 

1.389011    15 

40 

.0-1303 

1.55368 

.66800 

1.49:,00 

.09416 

1.44060 

.72034 

1.388-,'t     11 

47 

.04404 

1.55269 

.00902 

1.4947'2       .09459 

1.43970 

.72078 

1.387'38     13 

48 

.04446 

1.55170 

.00944 

.   1.40378  i     .(;:):>(  12 

1.43881 

.73122 

1.38053    12 

1!)     .ilUST 

1.55071 

.06980      1.49284    '    .0»r;i:> 

1.43792 

.72107 

1.38568 

11 

50 

.04528 

1.54972 

.67028     1.49190       .69588 

1.43703 

.72211 

1.38184     10 

51 

.04509 

1.54873 

.67071 

1.49097       .09631 

1.43614       .72255 

1.38399 

( 

52 

.04610 

1.54774 

.07113 

1.49003 

.69675 

1.43525        .72-21)11 

1.38314 

8 

53 

.04052 

1.54075 

.07155 

1.48909 

.69718 

1.43430 

.72344 

1.38229 

54 

.64693 

1.545rO 

.67197 

1.48810 

.69701 

1.48847 

.72388 

1.381  !•"> 

6 

55 

.64734 

1.54478 

.07239 

1.48722 

.09804 

1.43258 

.72432 

L38060 

5 

5b 

.64715 

1.54879 

!  67282 

1.48029 

.09847 

1.43109 

.72477 

1.37976 

i 

5? 

.64817 

1.54281 

.67324 

1.48536 

.09891 

1.43080 

.72521 

j 

56 

.04358 

1.54183 

.07300 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

5C 

,64399 

1.54085 

!    .07409 

1.483-19 

.09977 

1.42903 

!  72610 

1.37722 

-\ 

GC 

.04941 

1.53980 

.07451      1.48250 

.70021 

1.42815 

.72654 

1.87638 

0 

' 

Cotang  |    Tang 

Cotang  |    Tang 

Cotang 

Tang 

C<>tang|    Tang     j  ^ 

57° 

!           56° 

55°           ii           54° 

TABLE  XXVIII. -NATURAL  TANGENTS  AND  COTANGENTS. 


36°                       37°  • 

38° 

39° 

Tang 

Cotang  1  1  Tang 

Cotang 

Tang  |  Cotang 

Tang     Cotang 

0 

.72054 

1.37038 

.75355 

1.82704 

.78129 

1.27994 

.80978 

1.2:3490 

60 

1 

.72699 

1.37554 

.75401 

1.32024 

.78175 

1.27917 

.81027 

1.23416 

59 

8 

72748 

1.37470 

.75447 

1.32514 

.78222 

1.27841 

.81075 

1.2:3343 

58 

8 

!  72788 

1.37380 

.75492 

1.32464 

.78209 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384  i 

.7'8316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304  1 

.78863 

1.27611 

.81220 

1.23123 

55 

i; 

.72921 

1.371:34 

.75029 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

1 

.72966 

1.37050  i 

.75075 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36907 

.75721 

1.32004 

.  78504 

1.27382 

.81364 

1.22904 

52 

91   .73055 

1.86883 

.75707 

1.31984  ; 

.7'8551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.30800 

.75812 

1.81904 

.78588 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.30716  ! 

.75858 

1.31825 

.78645 

1.27153 

.81510 

.22685 

49 

12 

.73189      1.30(133 

.75901 

1.31745 

.78692 

1.27077 

.81558 

.22012 

48 

13:    .73231 

1.30549 

.75950 

1.31000 

.787'39 

1.27001 

.81606 

.22539 

47 

14  i  .73-78 

1  .  30400 

.75990 

1.31586 

.78786 

1.26925 

.81655 

.22407 

46 

1.-)   .73:333 

1.36383 

.70042 

1.31507  j 

.78834 

1.26849 

.81703 

.22394 

45 

16 

.73308 

1.36300 

.76088 

1.31427  i 

.78881 

1.26774 

.81752 

.22321 

44 

17 

.73413 

1.36217 

.76134 

1.31348  1 

.78928 

1.26698 

.81800 

.2224!) 

43 

18!   .73457 

1.36134 

.70180 

1.31269  1 

.78975 

1.26622  ' 

.81849 

.22176 

42 

19     .7350-3 

1.36051 

.7022(1 

1.31190  i 

.79022      1.26546 

.81898 

.22101 

41 

20     .73547 

1.85968 

.70272 

1.31110 

.79070 

1.26471 

.81946 

.22031 

40 

21     .73502 

1.35885 

.76318 

1.31031  1 

.79117 

1.26395 

.81995 

.21959 

39 

22 

.73037 

1.35802 

.76304 

1.30952  ' 

.79164 

1.26319 

.82044 

.21880 

38 

23 

.73081 

1.35719 

.76410 

1.30873 

.79212 

1.20244 

.82092 

.21814 

37 

:.'! 

.73720 

1.35037 

.70450 

1.30795 

.79259 

1.26169 

.82141 

.21742 

36 

25 

.73771 

1.35554 

.70502 

1.30716 

.79306 

1.26093 

.82190 

.21670 

35 

20 

.73816 

1.35472 

.76548 

1.30G37 

.79354 

1.26018 

.82238 

.21598 

34 

27 

.73861 

1.351389 

.78594 

1.30558 

.79401 

1.25943 

.82287 

.21526 

33 

28     .73!  «n  5 

1.35307 

.70040 

1.30480 

.79449 

1.25867 

.82336 

.21454 

32 

2!) 

.78951 

1.35224 

.76086 

1.30401 

.79498 

1.25792 

.82385 

.21382 

31 

80 

.73990 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.824134 

.21310 

30 

::i 

.74041 

1.35080 

.76779 

1.30244 

.79591 

1.25642 

.S24&3 

.21238 

29 

82 

.74086 

1.34978 

.76825 

1.30106 

.79639 

1.25507 

.82531 

.21166 

28 

88 

.74131 

1.34896 

.70871 

1.30087 

.79086 

1.25492 

.82580 

.21094 

27 

34 

.74170 

1.34814 

.70918 

1.30009 

179784 

1.25417 

.82629 

.21023 

S3 

35     .74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

.20951 

25 

86 

.74207 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

.20879 

24 

31 

.74312 

1.34508 

.77057 

1.29775 

.79877 

1.25193 

:82776 

.20808 

23 

88 

.74857 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

.20786 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

.20593 

20 

n 

.74492 

1.34242 

77242 

1.29463 

.80067 

1.24895 

.82972 

.20522 

19 

42 

.74538 

1.34160 

!  77289 

1.29385 

.80115 

1.24820 

.83022 

.20451 

18 

43     .7458:] 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

.20379 

17 

44     .7402S 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

.20308 

16 

45     .74(174 

1.83916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

.20287 

15 

46     .7471!) 

1.  .33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20100    1-! 

17 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48     .74N10 

1.33073 

.77568 

1.28919 

.80402 

1.24375 

.&3317 

1.20024 

12 

4!)     .718.V> 

1.33592 

.77015 

1.28842 

.80450 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77601 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

.19811' 

9 

552 

.74991 

1.33349 

.  77754 

1.28610 

.80594 

1.24079 

.88514 

.19740 

8 

53 

.75037 

1.33868 

.77801 

1.28533 

.80642 

1.24005 

.83564 

.19009 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

.19599 

6 

B5 

.75128 

1.33107 

.77895 

1  128379 

.807:38 

1.2:3858 

.83602 

.19528 

5 

6fl 

.75173 

1.83096 

.77941 

1.28302 

.80786 

1.23784 

.83712 

.19457 

4 

r>; 

.7521'.) 

1.32940 

.77988 

1.28225 

.808:34 

1.23710 

.88781 

.19387 

3 

58 

.75864 

1.328(15 

.78035 

1.28148    1    .80882 

1.23637 

.83811 

.19316 

2 

59 

.75310 

1.32785 

.79082 

1.28071 

.809:30 

1.23503 

.83860 

.19246 

1 

(,•. 

.75865 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

.19175 

0 

! 

Cbtang 

Tang      Cotang 

Tang      Cotang 

Tang 

Cotang  j    Tang 

53°           il           52°           ii           51° 

1           50°           1- 

407 


TABLE  XXVIII.— NATURAL  TANGENTS  AND  COTANGENTS. 


40°           „           41o           p        •  42o                       43o 

/ 

Tang   1  Cotang      Tang     Cotang  j    Tang     Cotang 

Tang     Cotang 

0 

.83910 

1.19175 

.80929      1.15037  ji    .90040      1.11061 

.9:5252      1.07237 

60 

1 

.83960 

1.19105 

.86980 

1  .  14969 

.90093  i  1.10996 

.93806 

1.07174 

59 

8 

.84009 

1.19035 

.87031 

1.14902 

.90146  I  1.10931 

.93300 

1.07112 

58 

8 

.B4059 

1.18964 

.87082 

1.14834 

.90199      1.10867 

.98415 

L.  07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

!  90251 

1.10802 

.93469 

1.00987 

56 

:, 

.84158 

1.18824 

.87184 

1.14699 

.90:304 

1.10737 

.93524 

1.00925 

55 

6 

.84208 

1.18754 

.87236 

1.11032 

.90357      1.10672 

.93578 

1.06862 

54 

ri 

.84258 

1.18684 

.87287 

1.14565 

.90410      1.10607 

.93033 

1.06800 

53 

H 

.81307 

1.18614 

.87338 

1.1  H'.IS 

.90463      1.10543 

.93088  ;  1.067:58 

52 

9 

.84357 

1.18544 

.87389  !  1.11430 

.90516      1.10478 

.9:',743  i  1.00676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569      1.10414 

i   .93797 

1.00613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621      1.10349 

.93852 

1.06551 

49 

(2 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

;   .93906 

1.06489 

48 

18 

.84556 

1.18264 

.87595 

1.14162 

.90727  i  1.10220 

.93961 

1.06427    47 

n 

.84606 

1.18194 

.87646      1.14095 

.90781  !  1.10150 

.94010 

1.00365    40 

15 

.84656 

1.18125 

.87698      1.1402S 

.90834  !  1.10091 

.94071 

1.00303    45 

16     .£1706 

1.18055 

.87749     1.13961 

.90887     1.10027 

.94125      1.00241    44 

17 

.84756 

1.17986 

.87801      1.13894 

.90940      1.0!H!03 

.94180      1.06179    43 

IS 

.84806 

1.17916 

.87852  i  1.13828 

.90993      1.09S99- 

.94235      1.00117    42 

1!) 

.84850 

1.17846 

.87904 

1.13761 

.91040      1  .09834 

.94290      1.06056  j41 

20 

.84906 

.87955 

1.13694 

.91099 

1.09770 

.94345      1.05994    40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400  :  1.05932    39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09042 

.94455      1.05870  138 

-.'.•! 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510      1.05809 

37 

23 

.85107 

1.17500 

.88162 

1.13428 

.91313      1.  09514 

.94565 

1.05747 

86 

as 

.85157 

1  .  17430 

.88214 

1.13361 

.91366      1.09450 

.94620 

1.05685 

85 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419      1.09380 

.94676 

1.05624    31 

27 

.85257 

1.17292 

.88317 

1.13223 

.9147'3 

1.09322 

.94731 

1.05562    33 

28 

.85308 

1.17223 

.88369 

1.13162 

.91526      1.09258 

.94786 

1.05501    32 

26 

.85358      1.17154 

.88421 

1.13096 

.91580      1.09195 

.94841      1.05489    31 

30 

.85408 

1.17085 

.88473 

1.13029 

.916*3 

1.09131 

.94896  i  1.05378    30 

31 

.85458 

1.17016 

.88684 

1.12963 

.91687 

1.09067 

.94952      1.05817 

80 

82 

.85509 

1.16947 

.88576 

1.12897 

.91740      1.09003 

.95007  i  1.05255    28 

88 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95002      1.05194    27 

34 

.85609 

1.16809 

.88680 

1.12765 

.91847 

1.08876 

.95118      1.05133    20 

85 

.85660 

1  .  16741 

.88732 

1.12690 

.91901      1.08813 

.95173 

1.05072  125 

86 

.85710 

1.16672 

.88784      1.12633 

.91955      1.08749 

.95£>9      1.05010 

24 

37 

.85761 

1.16603 

.88836      1.12567 

.92008      1.08080 

.95284 

1.04949 

23 

88 

.85811 

1.165:35 

.88888 

1.12501 

.92i;(>3      1.08622 

.95840 

1.04888 

22 

80 

.85862 

1.16466 

.88940 

1.12435 

.92116      1.08559 

.95395 

1.04827 

21 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170  |  1.08496 

.95451 

1.04766 

20 

4\ 

.85968 

1.16329 

.89045     1.12303 

.92224      1.08432 

.95500 

1.04705 

19 

43 

.86014 

1.16261 

.89097      1.12238 

.92277      1.088(51) 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149      1.12172 

.92331   j  1.08300 

.95618 

1.04583  J17 

44 

.86115 

I.llil24 

.89201 

1.12106 

.92385      1.08243 

.95673 

1.04522  Il6 

45 

,86166 

1.16056 

.89253      1.12041 

.92139      1.08179 

.95729 

1.01401 

15 

4G 

.86216 

1.15987 

.89306      1.11975 

.92493      1.08116 

.95785 

1.04401 

1! 

4; 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

18 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1  -07990 

.95897      1.04279 

12 

49  !   .803C.S 

1.15783 

.89463 

1  11778 

.92055      1.07927 

.95952 

1.04218 

11 

50     .80419 

1.15715 

.89515 

1.11713 

.92709     1.07864 

.96008 

1.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92703      1.07801 

.96064 

1.04097 

0 

53 

.86521 

1  .  15579 

.89620 

1.115B2 

.92817 

1.07738 

.96120 

1.04030 

s 

58 

.86572 

1.15511 

-.89672      1.11517 

.92P72 

1.07070 

.96176 

1.03976 

7 

M 

.86623 

1.15443 

.81)725      1.11452 

.92920 

1.07613 

.96232 

1.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387       .92980 

1.07550 

.96288 

1.03855 

r, 

56 

.867'25 

1.15308 

.89830 

1.11321 

.93034 

1.07'487 

.96344 

1.03791 

4 

57 

.86776 

1.15240 

.89883      1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

:,s 

.86827 

1.15172       .8993:. 

1.11191 

.93143 

1.07-362 

.96457 

1.03674 

2 

50 

.8687'8 

1.15104       .89988 

1.11126 

.93197 

1.07299 

.96513 

1.03013 

1 

60 

.86929 

1.15037    I   .90040      1.11061 

.  93252 

1.07237 

.96569 

1.03553 

0 

/ 

Cotang 

Tang     s  Cotang     Tang    j  Cotang 

Tang       Cotang      Tang 

, 

1           49°           !            48°                       47°            i           46° 

468 


TABLE  XXVin. -NATURAL  TANGENTS  AND  COTANGENTS. 


44° 

,  I       «*• 

I    r 

44° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355  (40     40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

.02295     39  |  41 

.98901 

1.01112 

19 

2 

.966S1 

1.03433 

58 

22 

.97813 

.02236 

38  1  42 

.98938 

.01053 

18 

3 
4 

.96738 

.98791 

1.03372 
1.03312 

s 

23 
24 

.97870 
.97927 

.02176 
.02117 

37  1  !  43 
36  II  44 

.99016 
.99073 

.00994 
.009% 

17 
16 

5 

.96850 

1.C3252 

55 

25 

.97984 

.02057 

35     45 

.99131 

:  .90876 

15 

6 

.96907 

1.03192 

54 

26 

.98041 

.01998 

34  i   46 

.99189 

.(-0818 

14 

'7 

.96963 

1.03132 

63 

27 

.98098 

.01939 

33 

47 

.99247 

.00759 

13 

8 

.97020 

1.03072 

52 

28 

.98155 

.01879 

48 

.99304 

.00701 

12 

9 

.97076 

1.0J3012 

51 

29 

.98213 

.01820 

81 

49 

.99362 

.00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

.01761 

90 

50 

.99420 

.00583 

10 

11 

.97189 

1.02892 

49 

51 

.98327 

.01702 

01) 

51 

.99478 

.00525 

9 

12 

.97246 

1.02832 

48 

82 

.98381 

.01642 

28 

52 

.99536 

.00467 

8 

18 

.97302 

1.02772 

47 

38 

.98441 

.01583 

271   53 

.99594 

.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

.01524 

26!   54 

.99652 

.00.350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

.01465 

25     55 

.99710 

.00291 

5 

16 

.97472 

1.02593 

44 

36 

.98613 

.01406 

24     56 

.99768 

:  .00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23  1  !  57 

.99826 

.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

:  .00116 

a 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

81 

59 

.99942 

.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

.00000 

o 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

45° 

45° 

45° 

• 

TABLE  XXIX.— NATURAL  VERSED  SINES  AND    EXTERNAL    SECANTS. 


/ 

0° 

1° 

1 

2° 

3° 

i 

Vers. 

Ex.  sec. 

i  Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.00000 

.00000 

.00015 

.00015 

.00061 

.0006T 

.00137" 

.00137 

0 

1 

.00000 

.00000 

.00016 

.00016 

.000(32 

.OODli2 

.00139 

00139   1 

2 

.00000 

.00000 

.00016 

.00016 

.00083 

.00063 

.00140 

.00140  I  2 

3 

.00000 

.00000 

.00017 

.00017 

[00064 

.00064 

.00142 

.00142  I  3 

4 

.00000 

.00000 

.00017 

.00017 

.00066 

.00065  ' 

.00143 

.00143 

4 

5 

.00000 

.o;jooo 

.00018 

.00018 

.mom; 

.000(><> 

.00145 

.00145 

5 

6 

.00000 

.00000 

.00018 

.00018  ! 

.00067 

.00067  •• 

.00146 

.00147 

6 

7 

.00000 

.00000 

.00019 

.00019 

.00068 

.00068  ! 

.00148 

-.00148 

7 

8 

.00000 

.00000 

.00020 

.a*)2o 

.00069 

.00069  : 

.00150 

.00150 

8 

9 

.00000 

.00000 

.00020 

.00020 

.00070 

.00070 

.00151 

.00151 

9 

10 

.00000 

.00000 

.00021 

.00021 

.00071 

.00072  I 

.00153 

.00153 

10 

11 

.00001 

.00001 

.00021 

.00021 

.00073 

.00073 

.00154 

.00155 

11 

12 

.00001 

.00001 

.00022 

.00022 

.00074 

.00074 

.00156 

.00156 

12 

13 

.00001 

.00001 

.00023 

.00023 

.00075 

.00075 

.00158 

.00158 

13 

14 

.00001 

.00001 

.00023 

.00023 

.00076 

.00076 

.00150 

.00159 

14 

15 

.00001 

.00001 

.00024 

.00024 

.00077 

.00077 

\00101 

.00161 

15 

16 

.00001 

.00001 

.00024 

.00024 

.00078 

.00078  I 

.00162 

.00163 

16 

17 

.00001 

.00001 

.00025 

.0002.) 

.00079 

.00079 

.001(54 

.OOKi  4 

17 

18 

.00001 

.00001 

.00026 

.00026 

.00081 

.00081 

.00106 

.00166 

18 

19 

.00002 

.00002 

.00026 

.00026 

.00082 

.00082 

.00168 

.00168 

19 

20 

.00002 

.00002 

.00027 

.00027 

.00083 

.00083 

.00109 

.00169 

20 

21 

.00002 

.00002 

.00028 

.00028 

.00084 

.00084 

.00171 

.00171 

21 

22 

.00002 

.00002 

.00023 

.00028 

.00085 

.00085 

.00173 

.00173 

22 

23 

.00002 

.00002 

.00029 

.00029 

.00037 

.00087  ' 

.00174 

.00175 

23 

24 

.00002 

.00002 

.00030 

.00030 

.00088 

.00088 

.00176 

.00176  24 

25 

.00003 

.00003 

.00031 

.00031 

.00089 

.00089  ! 

.00178 

.00178 

25 

26 

.00003 

.00003 

.00031 

.00031 

.00090 

.00090  ! 

.00179 

.00180 

26 

27 

.00003 

.00003 

.00032 

.00032 

.00091 

.00091 

.00181 

.00182 

27 

28 

.00003 

.00003 

.00033 

.00033 

.00093 

.00093  | 

.00183 

.00183 

28 

29 

.00004 

.00004 

.00034 

.00034 

.00094 

.00094  ! 

.00185 

.00185 

29 

30 

.00004 

.00004 

.00034 

.00034 

.00095 

.00095 

.00187 

.00187 

30 

31 

.00004 

.00004 

.00035 

.00035 

.00096 

.00097  i 

.00188 

.00189 

31 

32 

.00004 

.00004 

.00036 

.00036 

.00098 

.00098  | 

.00190 

.00190 

32 

.33 

.00005 

.00005 

.00037 

.00037 

.00099 

.00099  ] 

.00192 

.00192 

33 

34 

.00005 

.00005 

.00037 

.00037 

.00100 

.00100 

.00194 

.00194 

34 

35 

.00005 

.00005 

.00038 

.00038 

.00102 

.00102 

.00196 

.00196 

35 

36 

.00005 

.00005 

.00039 

.00039 

.00103 

.00103  ; 

.00197 

.00198 

36 

37 

.00006 

.00006 

.00040 

.00040 

.00104 

.00104  i 

.00199 

.00200 

37 

38 

.00006 

.00006 

.00041 

.00041 

.00106 

.00106  i 

.00201 

.00201 

38 

39 

.00006 

.00006 

.00041 

.00041 

.00107 

.00107  : 

.00203 

.00203 

40 

.00007- 

.00007 

.00042 

.00042 

.00108 

.00108  i 

.00205 

.00205 

40 

41 

.oooor 

.00007 

.00043 

.00043 

.00110 

.00110 

.00207 

.00207 

41 

42 

.00007 

.00007 

.00044 

.00044 

.00111 

.00111 

.00208 

.00209 

42 

43 

.00008 

.00008 

.00045 

.00045 

.00112 

.00113 

.00210 

.00211 

43 

44 

.00008 

.00008 

.00046 

.00046 

.00114 

.00114 

.00212 

.00213 

44 

45 

.00009 

.00009 

.00047 

.00047 

.00115 

.00115 

.00214 

.00215 

45 

46 

.00009 

.00009 

.00048 

.00048 

.00117 

.00117 

.002W5 

.00216 

46 

47 

.00009 

.00009 

.00048 

.00048 

.00118 

.00118 

.00218 

.00218 

47 

48 

.00010 

.00010 

.00049 

.00049 

.00119 

.00120 

.00220 

.00220 

48 

49 

.00010 

.00010 

.00050 

.00050 

.00121 

.00121 

.00222 

.00222 

49 

50 

.00011 

.00011 

.00051 

.00051 

.00122 

.00122 

.00224 

.00224 

50 

51 

.00011 

.00011 

.00052 

.00052 

.00124 

.00124 

.00226 

.00226 

51 

52 

.00011 

.00011 

.00053 

.00053 

.00125 

.00125 

.00228 

.00228 

52 

53 

.00012 

.00012 

.000.54 

.00054 

.00127 

.00127  i 

.00230 

.00230 

53 

54 

.00012 

.00012 

.00055 

.00055 

.00128 

.00128 

.00232 

.00232 

54 

55 

.00013 

.00013 

.00056 

.00056 

.00130 

.001:30 

.00234 

.00234 

55 

56 

.00013 

.00013 

.00057 

.00057 

.00131 

.00131 

.00236 

.00236 

56 

57 

.00014 

.00014 

.00058 

.00058 

.00133 

.00133 

.00238 

.00238 

57 

58 

.00014 

.00014 

.00059 

.00059 

.00134 

.001:34 

.00240 

.00240 

58 

59 

.00015 

.00015 

.00060 

.00060 

.00136 

.00136 

.00242 

.00242 

59 

60 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

.00244 

.00244 

60 

470 


TABLE  XXIX.— NATURAL  VERSED  SINES    AND  EXTERNAL   SECANTS. 


40 

5° 

6° 

7° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

I 
Vers. 

Ex.  sec.  | 

0 

.00244 

.00244 

.00381 

.00382 

.00548 

.00551 

.00745 

.00751   0 

1 

.00246 

.00846 

.00383 

.00385 

.00551 

.00554 

.00749 

.00755   1 

2 

.00248 

.00248 

.00386 

.00387 

.00554 

.00557 

.00752 

.corns  2 

3 

.00250 

.00250 

.00388 

.00390 

.00557 

.00560 

.00756 

.00762   3 

4 

.00253 

.00252 

!  00891 

.00392 

.00560 

.00563 

.00760 

.00765 

4 

5 

.00254 

.00-254 

.00393 

.00395 

.00563 

.00566 

.00763   .00769 

5 

6 

.00256 

.00257 

.00396 

.00397 

.00566 

.00569 

.00767 

.00773  !  6 

.00258 

.00259 

.00398 

.00400 

.00569 

.00573 

.00770 

.00776 

7 

8 

.00260 

.00261 

.00401   .00403 

.00572 

.00576 

.00774 

00780 

8 

9 

.00262 

.00263 

.00404   .00405 

.00576 

.00579 

.00778 

.00784 

9 

10 

.00264 

.00265 

.00406 

.00408  ; 

.00579 

.00582 

.00781 

.00787 

10 

11 

.00266 

.00267 

.00409 

.00411 

.00582 

.00585 

.00785 

.007'91 

11 

12 

.00269 

.00269 

.00412 

.00413 

.00585 

.00588 

.00789 

.00795 

12 

13 

.00271 

.00271 

.00414 

.00416 

.00588 

.00592 

.00792 

.00799 

13 

14 

.00273 

.00274 

.00417 

.00419 

.00591 

.00595 

.00796 

.00802 

14 

15 

.00275 

.00276 

.00420 

.00421 

.00994 

.  .00598 

.00800 

.0080(5 

15 

16 

.00277 

.00278 

.00422 

.00424 

.00598 

.00601 

.00803 

.00810 

16 

17 

.00279 

.00280 

.00425 

.00427 

.00601 

.00604  1 

.00807 

.00813 

17 

18 

.00281 

.00282 

.00438 

.00429 

.00(504 

.00(508  i 

.00811 

.00817 

18 

19 

.00284 

.00284 

.00430 

.00432 

.00607 

.00611  I 

.00814 

.00821 

19 

20 

.00286 

.00287 

.00433 

.00435 

.00610 

.00614 

.00818 

.00825 

20 

21 

.00288 

.00289 

.00436 

.00438 

.00614 

.00617 

.00822 

.00828 

21 

22 

.00290 

.00291 

.00438 

.00440 

.00617 

.00621  i 

.00825 

.00832 

22 

23 

.00293 

.00293 

.00441 

.00443 

.00620 

.00624 

.00829 

.00836 

23 

24 

.00295 

.00296 

.00114 

.00446 

.00623 

.00627 

.00833 

.00840 

24 

25 

.00297 

.00298 

.00447 

.00449 

.00626 

.00630 

.00837 

.00844 

25 

26 

.00299 

.00300 

.00449 

.00451 

.00630 

!  00684  \ 

.00840  1  .00848 

26 

27 

.00301 

.00302 

.00452 

.00454 

.00633 

.00637 

.00844  i  .00851 

27 

28 

.00304 

.00305 

.00455 

.00457 

.00636 

.00640  i 

.00848 

.00855 

28 

2!) 

.00309 

.00307 

.00458 

.00460 

.00640 

.00644  1 

.00852 

.ooa59 

29 

30 

.00308 

.00309 

.00460 

.00463 

.00643 

.00647 

.00856 

.00863 

30 

31 

.00311 

.00312 

.00463 

.00465 

.00646 

.00650 

.00859 

.00867 

31 

32 

.00313 

.00314 

.00466 

.00468 

.00(549 

.00654 

.00863 

.00871 

32 

33 

.00315 

.00316 

.00469 

.00471 

.00653 

.00657 

.00867 

.00875 

33 

34 

.00317 

.00318 

.00472 

.00474 

.00656 

.00660 

.00871 

.00878 

34 

86 

.00320 

.00321 

.00474 

.00-177 

.00659 

.00664 

.00875 

.008S2 

35 

88 

.00322 

.00323 

.00477 

.00480 

.00668 

.00667 

.00878 

.00886 

36 

37 

.00324 

.00326 

.00480 

.00482 

.00666 

.00671 

.00882 

.00890 

37 

38 

.00327 

.00328 

.00483 

.00485 

.00669 

.00674 

.00886 

.00894 

38 

39 

.00329 

.00330 

.00486 

.00488 

.00673 

.00677 

.00890 

.00898 

39 

40 

.00X32 

.00333 

.00489 

.00491 

.00676 

.00681 

.00894 

.00902 

40 

41 

.00334 

.00335 

.00492 

.00494 

.00680 

.00684 

.00898 

.00906 

41 

42 

.00336 

.00337 

.00494 

.00497 

.00683 

.00688 

.00902 

.00910 

42 

43 

.00339 

.00340 

.00497 

.00500 

.00686 

.00691 

.00906 

.00914 

43 

44 

.00311 

.00342 

.00500 

.00503 

.00690 

.00695 

.00909 

.00918 

44 

45 

.00343 

.00345 

.00503 

.00506 

.00693 

.00698 

.00913 

.00922 

45 

46 

.00346 

.00317 

.00506 

.00509 

.00697 

.00701 

.00917 

.00926 

46 

47 

.00348 

.00350 

.00509 

.00512 

.00700 

.00705 

.00921 

.00930 

47 

48 

.00351 

.00352 

.00512 

.00515 

.00703 

.00708 

.00925 

.00934 

48' 

49 

.00353 

.00354 

.00515 

.00518 

.00707 

.00712 

.00929 

.00938 

49 

50 

.00356 

.00357 

.00518 

.00521 

.00710 

.00715 

.00933 

.00942 

50 

51 

.00358 

.00359 

.00521 

.00524 

.00714 

.00719 

.00937 

.00946 

51 

52 

.00361 

.00362 

.00524 

.00527 

.00717 

.00722 

.00941 

.00950 

52 

53 

.00363 

.00364 

,00527 

.00530 

.00721 

.00726 

.00945 

.00954 

53 

54 

.00865 

.00367 

.00530 

.00533 

.00784 

.00730 

.00949 

.00958 

54 

55 

.00368 

.00369 

.00533 

.00536 

.00728 

.00733 

.00953 

.00962 

55 

56 

.00370 

.00372 

.00536 

.00539 

.00731 

.00737 

.00957 

.00966 

56 

57 

.00373 

.00374 

.00539 

.00512 

.00735 

.00740 

.00961 

.00970 

57 

58 

.00375 

.00377 

.00542 

.00545 

.00738 

.00744 

.00965 

.00975 

58 

59 

.00378 

.00379 

.00545 

.00518 

.00742 

.00747 

.00969 

.00979 

59 

60 

.00381 

.00382 

.00548 

.00551 

.00745 

.00751  1 

.00973 

.00983 

60 

TABLE   XXIX.-NATURAL   VERSED  SINES  AND  EXTERNAL    SECANTS. 


II            i             i            I 

/ 

8° 

9° 

10° 

11° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec.  :  Vers.  Ex.  sec.  | 

Vers. 

Ex.  sec. 

~ 

.00973 

.oooas  i  .01231 

.01247   .01519   .01543  j  .01837   .01872   0 

1 

.00977 

.00987   .01236 

.01251   .01524   .01548  ' 

.01843   .01877   1 

2 

.00981 

.00991 

.01240 

.01256  ;  .01529 

.03553 

.01848 

.01883 

2 

3 

.00985 

.00995  ! 

.01245 

.012(51   .015:34 

.01558 

.01854 

.01889 

3 

4 

.00989 

.00999 

.01249 

.01265   .01540 

.01564 

.01860 

.01895   4 

5 

.00994 

.01004  l 

.01254 

.01270   .01545 

.01569 

.01805 

.01901 

5 

6 

.00998 

.01008 

.01259 

.01275  !|  .01550 

.01574 

.01871 

.01906 

6 

7 

.01002 

.01012  ': 

.01263 

.01279   .01555 

.01579  1 

.01876 

.01912 

7 

8 

.01006 

.01016 

.01268 

.01284  !;  .()15(iO 

.01585 

.01882 

.01918 

8 

9 

.01010 

.01020   .01272 

.01289  i  .Ojr.li.l 

.01590   .01888 

.01924 

9 

10 

.01014 

.01024  ;  .01277 

.012-J4   .01570 

.01595 

.01893 

.01930 

10 

11 

.01018 

.01029  ! 

.01282 

.01298 

.01575 

.01601  ' 

.01809 

.01936 

11 

12 

.01022 

.01033 

.01286 

.01303 

.01580 

.01606   .01904 

.01941 

12 

13 

.01027 

.01037 

.01291 

.01308 

.01586 

.01611 

.01910 

.01947 

13 

14 

.01031 

.01041 

.01296 

.01313 

.01591 

.01616 

.01916 

.01953 

14 

15 

.01035 

.01046 

.01300 

.01318 

.01596 

.01622 

.01921 

.01959 

15 

16 

.01039 

.01050 

.01305 

.01323 

.01601 

.01627 

.01927 

.01965 

16 

17 

.01043 

.01054 

.01310 

.01327 

.01606 

.01633 

.01933 

.01971 

17 

18 

.01047 

.01059 

.01314 

.01332 

.01612 

.01638 

.01939 

.01977 

18 

19 

.01052 

.01063 

.01319 

.01337 

.01617 

.01643 

.01944 

.01983 

19 

20 

.01056 

.01067 

.01324 

.01342 

.01622 

.01649 

.01950 

.01989 

20 

21 

.01080 

.01071 

.01329 

.01346 

.01627 

.01654 

.01956 

.01995 

21 

22 

.01034 

.01076 

.01333 

.01351 

.01632 

.01659 

.01961 

.02001 

22 

23 

.01069 

.01030 

.01338   .01356 

.01638 

.01665 

.01%7 

.02007 

23 

24 

.01073 

.01034 

.01343 

.01361 

.01643 

.01670 

.01973 

.02013 

24 

25 

.01077 

.01089 

.01348 

.01366 

.01648 

.01676 

.01979 

.0201'.) 

25 

26 

.01081 

.01093  : 

.01352 

.01371 

.01653 

.01681 

.01984 

.02025 

26 

27 

.01036 

.01097 

.01357 

.01376 

.01659 

.01687 

.01990 

.02031 

27 

28 

.01090 

.01102   .01362 

.01381 

.01664 

.01692 

.01  !«.«5 

.020:37 

28 

29 

.01094 

.01106 

.01367 

.01386 

.01669 

.01698 

.02J02 

.02043 

29 

30 

.01098 

.01111  j 

.01371 

.01391 

.01675 

.01703 

.02008 

.02049 

30 

31 

.01103 

.01115  ! 

.01376 

.01395 

.01680 

.01709 

.02013 

.02055 

31 

32 

.01107   .01119 

.01381 

.01400 

.01685 

.01714 

.02019 

.02061 

82 

33 

.01111 

.01124 

.01386 

.01405 

.01690 

.01720 

.02025 

.02067 

83 

34 

.01116 

.01128 

.01391 

.01410 

.011)96 

.01725 

.02031 

.02073  34 

35 

.01120 

.01133  ; 

.01396 

.01415 

.01701 

.01731 

.02037 

.02079 

35 

36 

.01124 

.01137  i  .01400 

.01420 

.01701) 

.01736 

.02042 

.02085 

36 

37 

.01129 

.01142   .01405 

.01425 

.01712 

.01742 

.02048 

.02091  !  37 

38 

.01133 

.01146 

.01410 

.014:30 

.01717 

.01747 

.02054 

.02097 

38 

39 

.01137 

.01151  1 

.01415 

.01435 

.01723 

.01753 

.02050 

.02103 

89 

40 

.01142 

.01155 

.01420 

.01440 

.01728 

.01758 

.0-2066 

.02110 

40 

41 

.01146 

.01160 

.01425 

.01445 

.01733 

.01764 

.02072 

.02116  41 

42 

.01151 

.01164 

.01430 

.01450 

.01739  1  .01769 

.02078 

.02122 

42 

43 

.01155 

.01169 

.01435 

.01455 

.01744 

.01775 

.02084 

.02128 

43 

44 

.01159 

.01173 

.01439 

.01461 

.01750 

.01781 

.02090 

.02134 

44 

45 

.01164 

.01178 

.01444 

.01466 

.01755 

.01786 

.02095 

.02140 

45 

46 

.01168 

.01182 

.01449  !  .01471 

.01760 

.01792 

.02101 

.02146 

46 

47 

.01173 

.01187 

.01451  i  '01476 

.01766 

.01798   .02l<»7 

.02153 

47 

48 

.01177 

.01191 

.01451) 

.01481 

.01771 

.01803  !  .02113 

.02159 

48 

49 

.01182 

.01196 

.01404 

.0148(5 

.01777 

.OWK).   .0:2119 

.02165 

49 

50 

.01186 

.012CO 

.01461) 

.01491 

.01782 

.01815   .02125 

.02171 

50 

i 

51 

.01191 

.01205 

.01474 

.01496 

.01788 

.01820 

.02131   .02178 

51 

52 

.01195 

.0120!) 

.01479  1  .01501 

.01793 

.01826 

.02137 

.02184 

52 

53 

.01200 

.01214 

.01484 

.01608 

.'01799 

.018® 

.02143 

.02190 

53 

54 

.01204 

.01219 

.01489 

.01  5]  -3 

.01804 

.01837 

.02149 

.02196 

54 

55 

.01209 

.01223 

.01494 

.1)1517 

.01810 

.01843 

.02155 

.02203 

55 

56 

.01213 

.01228 

.014!)!) 

.01522 

.01815 

.01849 

.02161 

.02205)   56 

57 

.01218 

.01233 

.01504 

.01527 

.01821 

.01854 

,02167 

.02215   57 

58 

.01222 

.01237 

.015(11) 

.01532 

.01826 

.01860 

.02173 

.02221   58 

59 

.01227 

.01242 

.01514 

.01537 

.01882 

.018(56   .02179 

.02228  59 

60 

.01231 

.01247  i 

.01519 

.01543 

.01837 

.0|S7v!   .U-MS5   -02234   60 

472 


TABLE    XXIX.— NATURAL  VERSED  SINES  AND  EXTERNAL    SECANTS. 


/ 

12°         13° 

14° 

15° 

/ 

Yers.  Ex.  sec.  1  Vers.  Ex.  sec. 

II      j 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.02185 

.02234 

.02563   .026:30 

.02970 

.03061 

.03407 

.03528 

0 

1 

.02191 

.02240 

.02570 

.02637 

.02977 

.OJ3069 

.03415 

.03536 

1 

2 

.02197 

.02247 

.02576 

.02644 

.02985 

.03076 

.03422 

.03544 

2 

3 

.0220.3 

.02253 

.02583 

.02651 

.02992 

.03084  ;  .03430 

.03552 

3 

4 

.02210 

.02259 

.02589 

.02658 

.02999 

.03091 

.03438 

.03560 

4 

5 

.02216 

.022(56 

.02596 

.02665 

.03006 

.03099  i 

.03445 

.03568 

5 

6 

.02222 

.02272 

.02602 

.02672 

.03013 

.03106  ! 

.03453 

.03576 

6 

7 

.02228 

.02279 

.02609 

.02679 

.03020 

.03114 

.03460 

.03584 

7 

8 

.02234 

.02285 

.02616 

.C2686 

.03027 

.03121 

.03468 

.03592 

8 

0 

.02240 

.02291 

.02622 

.02693 

.03034 

.03129 

.03476 

.03601 

9 

10 

.02246 

.02298 

.02629 

.02700 

.  .03041 

.03137 

.03483 

.03609 

10 

11 

.02252 

.02304 

.02635 

.02707 

.03048 

.03144 

.03491 

.03617 

11 

12 

.03268 

.02311 

.02642 

.02714 

.03055 

.03152  , 

.03498 

.03625 

12 

13 

.02205 

.02317 

.02649 

.02721 

.03063 

.03159  ' 

.03506 

.03633 

13 

14 

.02271 

.02323 

.02655 

.02728 

.03070 

.03167   .03514 

.03642 

14 

15 

.02277 

.023:30 

.02662 

.02735 

.03077 

.03175 

.03521 

.03650 

15 

16 

.02283 

.02:336 

.02669 

.02742 

.03084 

.03182 

.03529 

.03658 

16 

17 

.02289 

.02343 

.02675 

.02749 

.03091 

.03190 

.0&537 

.03666 

17 

18 

.02295 

.02:349 

.02682 

.02756 

.03098 

.03198 

.03544 

.03674 

18 

19 

.02302 

.02356 

.02689 

.02763 

.03106 

.03205 

.03552 

.03683 

19 

20 

.02308 

.02362 

.02696 

.02770 

.03113 

.03213 

.03560 

.03691 

20 

21 

.02314 

.02369 

.02702 

.02777 

.03120 

.03221 

.03567 

.03699 

21 

22 

.02320 

.02375 

.02709 

.02784 

.03127 

.03228 

.03575 

.03708 

22 

23 

.02327 

.02382 

.02716 

.02791 

.03134 

.03236 

.03583 

.03716 

23 

24 

.02333 

.02388 

.02722 

.02799 

.03142 

.03244 

.03590 

.03724 

24 

25 

.02339 

.02395 

.02729 

.02806 

.03149 

.03251 

.03598 

.03732 

25 

2(3 

.02345 

.(2402 

.02736 

.02813 

.03156 

.03259 

.03606 

.03741 

26 

27 

.02352 

.02408 

.02743 

.02820 

.03163 

.03267 

.03614 

.03749 

27 

28 

.02358 

.02415 

.02749 

.02827 

.03171 

.03275 

.03621 

.03758 

28 

29 

.02364 

.02421  I  .02756 

.02834 

.03178 

.03282 

.03629 

.03766 

29 

30 

.02370 

.02428 

i  .02763 

.02842 

.03185 

.03290 

.03637 

.03774 

30 

31 

.02377 

.02435 

.02770 

.02849 

.03193 

.03298 

.03645 

.03783 

31 

32 

.02383 

.02441 

.02777 

.02856 

.03200 

.03306 

.03653 

.03791 

32 

33 

.02389   .02448 

.02783 

.02863 

.03207 

.03313 

.03660 

.03799 

33 

34 

.02396 

.02454 

.02790 

.02870 

.03214 

.03321 

.03668 

.03808 

34 

35 

.02402 

.02461 

.02797 

.02878 

.03222 

.03329 

.03676 

.03810 

35 

36 

.02408 

.02468 

!  .02804 

.02885 

.03229 

.03337 

.03684 

.03825 

36 

37 

.02415 

.02474 

.02811 

.02892 

.03236 

.03345 

.03692 

.03833 

37 

38 

.02421 

.02481 

.02818 

.02899 

.03244 

.03353 

.03699 

.03842 

38 

39 

.02427 

.02488 

.02824 

.02907 

.03251 

.03360 

.03707 

.03850 

39 

40 

.02434 

.02494 

.02831 

.02914 

.03258 

.03368 

.03715 

.03858 

40 

41 

.02440 

.02501 

.02838 

.02921 

.03266 

.03376 

.03723 

.03867 

41 

42 

.02447 

.02508 

.02845 

.02928 

.03273 

.03384 

.03731 

.03875 

42 

43 

.02453 

.02515 

.02852 

.02936 

.03281 

.03392 

,03739 

.03884 

43 

44 

.02459 

.02521 

.02859 

.02943 

.03288 

.03400 

.03747 

.03892 

44 

45 

.02466 

.02528 

.02866 

.02950 

.03295 

.03408 

.03754 

.03901 

45 

46 

.02472 

.025:35 

.02873 

.02958 

.03303 

.03416 

.03762 

.03909 

46 

47 

.02479 

.02542 

.02880 

.02965 

.03310 

.03424 

.03770 

.03918 

47 

48 

.02485 

.02548 

.02887 

.02972 

.03318 

.0:3432 

.03778 

.03927 

48 

49 

.02492 

.02555 

.02894 

.02980 

.03325 

.03439 

.03786 

.03935 

49 

50 

.0249S 

.025U2  ' 

.02900 

.02987 

.03333 

.03447 

.03794 

.03944 

50 

51 

.02504 

.02509 

.02907 

.02994 

.03340 

.03455 

.03802 

.03952 

51 

;V2   .02511 

.02576 

.02914 

.03002 

1  .0:3347 

.03463 

.03810 

.03!)<;i 

52 

53 

.02517 

.02582 

.02921 

.03009 

.0:3355 

.03471 

.03818 

.03969 

53 

54 

.02524 

.02589 

.02928 

.03017 

.03362 

.03479  : 

.03826 

.03978 

54 

55 

.02530 

.02596 

.02935 

.03024 

.03370 

.03487 

.03834 

.03987 

55 

56 

.02537 

.02003 

.02942 

.03032 

.03377 

.03495 

.03842 

.03995 

56 

57 

.02543 

.02610 

.02949 

.03039 

.03385 

.03503 

.03a50 

.04004 

57 

58 

.02550 

.02617 

.02956 

.03046 

.0,3392 

.03512 

.08858 

.04013 

58 

59 

.02556 

.02624 

.02963 

.03054 

.03400 

.03520 

.03866 

.04021 

59 

60 

.02563 

.02630 

.02970 

.03061 

.03407 

.03528 

.03874 

.04030 

60 

TABLE   XXIX.— NATURAL  VERSED  SINES    AND  EXTERNAL   SECANTS. 


/ 

K 

>° 

1-3 

"   i 

** 

;° 

1! 

)" 

/ 

Vers. 

Ex.  sec.  ; 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.03874 

.04030 

.04370 

.04569 

.04894 

.05146 

.05448 

.05762 

0 

i 

.03882 

.04039 

.04378 

.04578 

.04903 

.05156  • 

.05458 

.05773 

1 

2 

.03890 

.04047 

.04387 

.04588  i 

.04912 

.05160 

.05467 

.05783 

2 

3 

.03898 

.04056 

.04395 

.04597 

.04921 

.05176  i 

.05477 

.05794 

3 

4 

.03906 

.04065  i 

.04404 

.04606 

.04930 

.05186 

.06486 

.05805 

4 

5 

.03914 

.0-1073  ; 

.04412 

.04616 

.04939 

.05196 

.054% 

.05815 

5 

6 

.03922 

.04082  j 

.04421 

.04625 

.04948 

.05206 

.05505 

.0,1S20 

6 

.03930 

.04091 

.04429 

.04635  ! 

.04957 

.05216 

.05515 

.05830 

7 

8 

.03938 

.04100 

.04438 

.04644  i 

.04967 

.05226 

.055:24 

.05847 

8 

9 

.03946 

.04108 

.04446 

.04653 

.04976 

.05230 

.05534 

.05868 

9 

10 

.03954 

.04117 

.04455 

.04663 

.04985 

.05246 

.05543 

.05809 

10 

11 

.03963 

.04126  '• 

.04464 

.04672 

.04994 

.05256 

.05553 

.05879 

11 

12 

.03971 

.04135 

.04472 

.04082 

.05003 

.05266  • 

.05502 

.05890 

12 

13 

.03979 

.04144  i 

.04481 

.04091 

.05012 

.05276 

.05572 

.05901 

13 

14 

.03987 

.04152 

.04489 

.04700 

.05021 

.05286 

.05582 

.05911 

14 

15 

.03995 

.04161 

.04498 

.04710 

.05030 

.05297 

-.05591 

.05922 

15 

16 

.04003 

.04170  ; 

.04507 

.04719 

.05039 

.05307 

.05(501 

.05933 

16 

17 

.04011 

.04179  ! 

.04515 

.04729 

.05048 

.05317  : 

.05010 

.05944 

17 

18 

.04019 

.04188  I 

.04524 

.04738 

.05057 

.05327 

.05020 

.05955 

18 

19 

.04028 

.04197  i 

.04533 

.04748 

.05007 

.05337 

.05630 

.05965 

19 

20 

.04036 

.04206  ] 

.04541 

.04757 

.05076 

.05347 

.05639 

.05976 

20 

21 

.04044 

.04214 

.04550 

.04767 

!  .05085 

.05357 

.05649 

.05987 

21 

22 

.04052 

.04-^3 

.04559 

.04776 

,  .05094 

.05367 

.05058 

.05998 

22 

23 

.04060 

.04232 

.04567 

.04786 

1  .05103 

.05378 

1  .05008 

.00009 

23 

24 

.04069 

.04241 

.04576 

.04795 

.05112 

.05388 

.05078 

.06020 

24 

25 

.04077 

.04250  i 

.04585 

.04805 

.05122 

.05398 

.05687 

.06030 

25 

20 

.04085 

.04259 

.04593 

.04815 

.05131 

.05408 

.05697 

.06041 

26 

27 

.04093 

.04268 

.04602 

.04824 

.05140 

.05418 

.05707 

.00052 

27 

•28 

.04102 

.04277 

.04611 

.04834 

.05149 

.05429 

.05710 

.00003 

28 

29 

.04110 

.04286 

.04620 

.04843 

|  .05158 

.05439  i 

.05726 

.06074 

29 

30 

.04118 

.04295 

.04628 

.04853 

I  .05168 

.05449 

.05736 

.06085 

30 

31 

.04126 

.04304 

.04637 

.04863 

.05177 

.05460 

.05746 

.00096 

31 

32 

.04135 

.04313 

.04646 

.04872 

.05186 

.05470 

.05755 

.00107 

32 

33 

.04143 

.04322 

.04655 

.04882 

.05195 

.05480 

.05705 

.06118 

33 

34 

.04151 

.04:331 

.04663 

.04891 

.05205 

.05490  ' 

.05775 

.00129 

34 

35 

.04159 

.04340 

.04672 

.04901 

.05214 

.05501  ' 

.05785 

.00140 

35 

36 

.04168 

.04349 

.04681 

.04911 

.05223 

.05511 

.05794 

.00151 

36 

37 

.04176 

.04358 

.04690 

.04920 

.05232 

.05521 

.05S04 

.00102 

37 

38 

.04184 

.04367 

.04699 

.04930 

.05242 

.05532 

.05814 

.00173 

38 

39 

.04193 

.04376  | 

.04707 

.04940 

.05251 

.(5542 

.05824 

.00184 

39 

40 

.04201 

.04385 

.04716 

.04951) 

.05260 

.05552 

.'05833 

.00195 

40 

41 

.04209 

.04394 

.04725 

.04959 

.05270 

.05563 

.05843 

.06206 

41 

42 

.04218 

.04403 

.04734 

.04909 

.05279 

.05573 

.05853 

.06217 

42 

43 

.04226 

.04413 

.04743 

.04979 

.05288 

.05584 

.05803 

.08228 

43 

44 

.04234 

.04422 

.04752 

.04989 

.05298 

.05594  : 

.05873 

.06239 

44 

45 

.04243 

.04431 

.04760 

.04998 

.05307 

.05604 

.05HK2 

.00250 

45 

46 

.04251 

.04440 

.04769 

.05008 

.05316 

.05615 

.05892 

.00061 

46 

47 

.04260 

.04449 

.04778 

.05018  i 

.05326 

.05625 

.05902 

.00272 

47 

48 

.04268 

.04458 

.04787 

.05028  | 

.05335 

.05630 

.05912 

.00283 

48 

49 

.04276 

.04468  ' 

.04796 

.05038  i 

.05344 

.05040 

.05922 

.OIW95 

49 

50 

.04285 

.04477 

.04805 

.05047 

.05354 

.05657 

.05932 

.06306 

50 

51 

.04293 

.04486 

.04814 

.05057 

.05363 

.05667 

.05942 

.06317 

51 

52 

.04302 

.04495 

.04823 

.05007 

.05373 

.05678 

.05951 

.06828 

52 

53 

.04310 

.04504 

.04832 

.05077 

.05382 

.05688 

.059(51 

!06389 

53 

54 

.04319 

.04514 

.04841 

.05087 

.05391 

.05699 

.05971 

.06350 

54 

55 

.04327 

.04523 

.04850 

.05097 

.05401 

.05709 

.05981 

.0(5362 

55 

56 

.04336 

.04532 

.04858 

.05107  i 

.05410 

.05720 

.05991 

.00373 

56 

57 

.04344 

.04541 

.04867 

.05116 

.05420 

.05730 

.0(5001 

.06384 

57 

58 

.04353 

.04551  j 

.04876 

.05126 

.0.5429 

.05741 

.00011 

.(Xvrfw 

58 

59 

.04361 

.04560 

.04885 

.05136 

.05439 

.05751 

.00021 

.00407 

59 

60 

.04370 

.04569 

.04894 

.05146  i 

.05448 

.05762 

.00031 

.06418 

60 

474 


TABLE    XXIX.— NATURAL  VERSED    SINES  AND  EXTERNAL    SECANTS. 


2( 

>° 

2] 

1° 

22 

22 

0 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.06031 

.06418 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

0 

1 

.06041 

.06429 

.06652 

.07126 

-  .07293 

.07866 

.07961 

.08649 

1 

2 

.06051 

.06440 

.06663 

.071:38 

.07-303 

.07879 

.07972 

.08663 

2 

3 

.06061 

.06452 

.06673 

.07150 

.07314 

.07892 

.07984 

.08676 

3 

4 

.06071 

.06463 

.08684 

.07162  | 

.07325 

.07904 

.07995 

.08690 

4 

5 

.06081 

.06474 

.06(394 

.07174 

.07336 

.07917 

.08006 

.08703 

5 

6 

.06091 

.06486 

.06705 

.07186 

.07347 

.07930 

.08018 

.08717 

6 

7 

.06101 

.06497 

.06715 

.07199 

i  .07358 

.07943 

.08029 

.08730 

7 

8 

.06111 

.06508 

.06726 

.07211 

.07369 

.07955 

.08041 

.08744 

8 

9 

.06121 

.065.20 

.06736 

.07*23 

.07380 

.07968 

.08052 

.08757 

9 

10 

.00131 

.08531 

.06747 

.07235 

.07391 

.07981 

.08064 

.08771 

10 

11 

.06141 

.06542 

.06757 

.07247 

.07402 

.07994 

.08075 

.08784 

11 

12 

.00151 

.08364 

.06768 

.07259 

.07413 

.08006 

.08086 

.08798 

12 

13 

.06161 

.06565 

.06778 

.07271 

.07424 

.08019 

.08098 

.08811 

13 

14 

.06171 

.06577 

.00789 

.07283 

.07435 

.08032 

.08109 

.08825 

14 

15 

.06181 

.06588 

.06799 

.07295 

.07446 

.08045 

.08121 

.08839 

15 

16 

.06191 

.08600 

.06810 

.07307 

.07457 

.08058 

.08132 

.08852 

16 

IT 

.06201 

.06611 

.06820 

.07320 

.07468 

.08071 

.08144 

.08866 

17 

18 

.06211 

.08622 

.06831 

.07332 

.07479 

.08084 

.08155 

.08880 

18 

19 

.06221 

.05634 

.  0(5841 

.07344 

.07490 

.08097 

.08167 

.08893 

19 

20 

.06231 

.06645 

.06852 

.07356 

.07501 

.08109 

.08178 

.08907 

20 

21 

.08341 

.06657 

.06863 

.07368 

.07512 

.08122 

.08190 

.08921 

21 

22 

.08252 

.08668 

.06873 

.07380 

.07523 

.08135 

.08201 

.08934 

22 

23 

.06262 

.08680 

.068,84 

.07393 

.07534 

.08148 

.08213 

.08948 

23 

24 

.06272 

.08891 

[08894 

.07405 

.07545 

.08161 

.08225 

.08962 

24 

25 

.06282 

.087'03 

.06005 

.07417 

.07556 

.08174 

.08236 

.08975 

25 

26 

.06292 

.06715 

.06916 

.07429 

.07568 

.08187 

.08248 

.08989 

26 

27 

.06302 

.0372(i 

i  .06926 

.07442 

.07579 

.08200 

.08259 

.09003 

27 

28 

.08312 

.06738 

i  .06937 

.07454 

.07590 

.08213 

.08271 

.09017 

28 

29 

.06323 

.06749 

i  .06948 

.07466 

i  .07601 

.08226 

.08282 

.09030 

29 

30 

.06333 

.03761 

1  .06958 

.07479 

.07612 

.08239 

.08294 

.09044 

30 

31 

.06343 

.03773 

.06969 

.07491 

.07623 

.08252 

.08306 

.09058 

31 

32 

.06-553 

.06784 

.06980 

.07503 

.07'634 

.08265 

.08317 

.09072 

32 

33 

.06363 

.08796 

.06890 

.07516 

.07645 

.08278 

.08329 

.09086 

33 

34 

.015374 

.0680?  i 

!  .070J1 

.07528 

.07657 

.08291 

.08340 

.09099 

34 

35 

.06:384 

.06819 

i  .07012 

.07540 

.07668 

.08805 

.08352 

.09113 

35 

36 

.06394 

.06831 

.07022 

.Or553 

.07679 

.08318 

.08364 

.09127 

36 

37 

.06104 

.06843 

.07U-J3 

.07565 

.07690 

.08331 

.08375 

.09141 

37 

33 

.06415 

.08854 

.07044 

.07578 

.07701 

.08344 

.08387 

.09155 

38 

39 

.06125 

.06866 

!  .07055 

.07590 

.07713 

.08357 

.08399 

.09169 

39 

40 

.06435 

.06878 

.07085 

.07602 

.07724 

.08370 

.08410 

.09183 

40 

41 

.06145 

.06889 

.07076 

.07615 

.077'35 

.08383 

.08422 

.09197 

41 

42 

.08456 

.06901  i 

i  .07087 

.07627 

.07746 

.08397 

.08484 

.09211 

42 

43 

.06466 

.06913  i 

.07098 

.07640 

i  .07757 

.08410 

.084,15 

.09224 

43 

44 

.06476 

.06925 

.07108 

.07652 

.07769 

.08423 

.08457 

.09238 

44 

45 

.06486 

.06936 

.07119 

.07665 

!  .07780 

.08436 

.08469 

.09252 

45 

40 

.08497 

.08948 

.07130 

.07677 

.07791 

.03449 

.08481 

.09266 

46 

47 

.05507 

.08980 

.07141 

.07690 

!  .07-802 

.08463 

.08492 

.09280 

47 

48 

.06517 

.08972 

.07151 

.07702 

.07814 

.08476 

.08504 

.09294 

48 

49 

.08528 

.03984  ! 

.07162 

.07715 

.07825 

.08489 

.08516 

.09308 

49 

50 

.08538 

.06995  : 

.07173 

.07727 

.07836 

.08503 

.08528 

.09323 

50 

51 

.06548 

.07007  i 

.07184 

.07740 

.07848 

.08516 

.08539 

.09&37 

51 

52 

.08559 

.o;-oi9 

.07195 

.07752 

.07859 

.08529 

.08551 

.09351 

52 

53 

.06569 

.07031 

.07206 

.07765 

i  .07'870 

.08542  1 

.08563 

.093(55 

53 

54 

.06580 

.07043 

.07216 

.0777-8 

.07881 

.0855(5 

.08575 

.09379 

54 

55 

.06590 

.07055 

.07227 

.07790 

.07893 

!  08569 

.08586 

.09393 

55 

56 

.06600 

.07087 

.07238 

.07803 

i  .07904 

.08582 

.08598 

.09407 

56 

57 

.06611 

.07079 

.07249 

.07810 

i  .07915 

.08596 

.08610 

.09421 

57 

58 

.06621 

.07091 

.07260 

.07828 

.07927 

.08609 

.08622 

.09435 

58 

59 

.06632 

.07103 

.07271 

.07841 

.07938 

.08623  i 

.08634 

.09449 

59 

60 

.06642 

.07115 

.07282 

.07853 

.0795.) 

.08636  ! 

.08645 

.09464 

60 

475 


TABLE   XXIX.— NATURAL  VERSED    SINES    AND  EXTERNAL   SECANTS 


24° 

25° 

26° 

27° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.08645 

.09464  I  .09369 

.10:338 

.10121 

.11260 

.10899 

.1SS33 

0 

1 

.08657 

.09478 

I  .09382 

.10353- 

.10133 

.11276 

.10913 

.12249 

1 

2 

.08669 

.09492 

.09394 

.  10368 

.10146 

.11292 

.10926 

.12206 

2 

3 

.08681 

.09506 

.09406 

.10:383 

.10159 

.11308 

.10939 

.12283 

8 

4 

.08693 

.09520 

.09418 

.10398 

.10172 

.11323 

.10952 

.12291) 

4 

5 

.08705 

.09535 

.09431 

.10413 

.10184 

.11339 

.10965 

.12316 

5 

6 

.08717 

.09549 

.09443 

.10428 

.10197 

.11355 

.10979 

.12333 

6 

7' 

.08728 

.09563 

.09455 

.10443 

.10210 

.11371 

.10992 

.12349 

7 

8 

.08740 

.09577 

.09468 

.10458 

.10223 

.11387 

.11005 

.12366 

8 

9 

.08752 

.09592 

.09480 

.10473 

.10236 

.11403 

.11019 

.12383 

9 

10 

.08764 

.09606 

.09493 

.10488 

.10248 

.11419 

j  .11032 

.12400 

10 

11 

.08776 

.09620 

.09505 

.10503 

.10201 

.11435 

.11045 

.12416 

11 

12 

.08788 

.09635 

.09517 

.10518 

.10274 

.11451 

.11058 

.12433 

12 

13 

.08800 

.09649 

.09530 

.10533 

.10287 

.11467 

.11072 

.12450 

13 

14 

.08812 

.09663 

.09542 

.10549 

.10300 

.11483 

.11085 

.12467 

M 

15 

.08824   .09678 

.09554 

.10564 

.10313 

.11499^ 

.11098 

.12484 

15 

16 

.08836   .09692 

.09567 

.10579 

.10326 

.11515 

.11112 

.12501 

16 

17 

.08848   .09707 

.09579 

.10594 

.10338 

.11531 

.11125 

.12518 

17 

18 

.08860 

.09721 

.09592 

.10609 

.10351 

.11547 

.11138 

.12534 

18 

19 

.08872 

.09735 

.09604 

.10625 

.10364 

.11563 

.11152 

.12551 

19 

20 

.08884 

.09750 

.09617 

.10640 

.10377 

.11579 

.11165 

.12568 

20 

21 

.08896 

.09764 

.09629 

.10655 

.10390 

.11595 

.11178 

.12585 

21 

22 

.08908 

.09779 

.09642 

.10670 

.10403 

.11611 

.11192 

.12602 

22 

23 

.08920 

.09793 

.09654 

.10686 

.10416 

.11627 

.11205 

.12619 

23 

24 

.08932 

.09808 

.09666 

.10701 

.10429 

.11643 

.11218 

.12636 

24 

25 

.08944 

.09822 

.09679 

.10716 

.10442 

.11659 

.11232 

.12653 

25 

26 

.08956 

.09837 

.09691 

.10731 

.10455 

.11675 

.11245 

.12670 

20 

27 

.08968 

.09851  ij  .097'04 

.10747 

.10468 

.11691 

.11259 

.12687 

27 

28 

.03980 

.09866   .09716- 

.10762 

.10481 

.11708 

.11272 

.12704 

28 

29 

.08992 

.09880  !  .09729 

.10777 

.10494 

.11724 

.11285 

.12721 

29 

30 

.09004 

.09895   .09741 

.10793 

.10507 

.11740 

.11299 

.12738 

30 

31 

.09016 

.09909   .09754 

.10808 

.10520 

.11756 

.11312 

.12755 

31 

32 

.09028 

.09924  ||  .09767 

.10824 

.10533 

.1177'2 

i  .11326 

12772 

32 

33 

.09040 

.09939 

.0977'9 

.10839 

.10546 

.11789 

.11339 

.12789 

33 

34 

.09052 

.09953 

.09792 

.10854 

.10559 

.11805 

.11353 

.12807 

34 

35 

.09064 

.09968 

.09804 

.10870  i  .10572 

.11821 

.11366 

.12824 

35 

36 

.09076 

.09982 

.09817 

.10885   .10585 

.11838 

.11380 

.12841 

36 

37 

.09089 

.09997 

.09829 

.10901 

.10598 

.11854 

.11393 

.12858 

37 

38 

.09101 

.10012 

.09842 

.10916 

.10611 

.11870 

.11407 

.12875 

38 

39 

.09113 

.10026 

.09854 

.10932 

.10624 

.11886 

.11420 

.12892 

30 

40 

.C9125 

.10041 

.09867 

.10947 

.10037 

.11903 

.11434 

.12910 

40 

41 

.09137 

.10055 

.09880 

.10963 

.10650 

.11919 

.11447 

.12927 

41 

42 

.09149 

.10071 

.09892 

.10978 

.10663 

.  1  1936 

.11461 

.12944 

42 

43 

.09161 

.10085 

.09905 

.10994 

.10676 

.11952 

.11474 

.12961 

43 

44 

.09174 

.10100 

.09918 

.11009 

.10689 

.11968 

i  .11488 

.12979 

44 

45 

.09186 

.10115 

.09930 

.11025 

.10703 

.11985 

.11501 

.12996 

45 

4(J 

.09198 

.10130 

.09943 

.11041 

.10715 

.12001 

;  .11515 

.13013 

46 

47 

.09210 

.10144 

.09955 

.11056 

.10728 

.12018 

.11528 

.13031 

47 

48 

.09232 

.10159 

.09968 

.11072 

.10741 

.12034 

.11542 

.13048 

48 

49 

.09234 

.10174 

.09981 

.11087 

.1075o 

.12051 

.11558 

.13065 

49 

50 

.09247 

.10189 

.09993 

.11103 

.10768 

.12067 

.11509 

.13083 

50 

51 

.09259 

.10204 

.10006 

.11119 

.10781 

.12084 

.11583 

.13100 

51 

52 

.09271 

.10218 

.10019 

.11134 

.10794 

.12100 

.11596 

.13117 

52 

53 

.09283 

.10233 

.10032 

.11150 

.10807 

.12117 

.11610 

.13135 

53 

54 

.09296 

.10248 

.10044 

.11166 

.10820 

.12133 

.11623 

.13152 

54 

55 

.09308 

.10263 

.10057 

.11181 

.10833 

.12150 

.11637 

.13170 

55 

56 

.09320 

.10278 

.10070 

.11197 

.10847 

.12166 

.11651 

.13187 

56 

57 

.09332 

.10293 

.10082 

.11213 

.10860 

.12183 

.11664 

.13205 

57 

58 

.09345 

.10308 

.10095 

.11229 

.10873 

.12199 

.11678 

.132-22 

58 

59 

.09357 

.10323 

.10108 

.11244 

.10886 

.12216 

.11692 

.13240 

59 

60 

.09369   .10:338 

.10121 

.11260  ! 

.10899 

.  12233 

.11705 

.13257 

60 

476 


TABLE  XXIX.— NATURAL   VERSED    SINES    AND  EXTERNAL  SECANTS. 


2 

5° 

2 

9= 

3( 

>° 

3] 

L° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.11705 

.13257 

.12538 

.14335 

.13397 

.15470 

.14283 

.16663 

0 

1 

.11719 

.13275 

.12552 

.14354 

.13412 

.15489 

.14298 

.16684 

1 

2 

.11733 

.13292 

.12566 

.14372 

.13427 

.15509 

.14313 

.16704 

2 

3 

.11746 

.13310 

.12580 

.14391 

.13441 

.15528 

.14328 

.16725 

3 

4 

.11760 

.1.3327 

.12595 

.14409 

.13450 

.  15548 

.14343 

.16745 

4 

5 

.11774 

.13345 

.12609 

.14428 

.13470 

.15567 

.14358 

.16766 

5 

0 

.11787 

.13362 

.12623 

.14446 

.13485 

.15587 

.14373 

.16786 

6 

7 

.11801 

.13380 

.12637 

.14465 

.13499 

.15606 

.14388 

~.  16800 

7 

8 

.11815 

.13398 

.12651 

.14483 

.13514 

.15626 

.14403 

.16827 

8 

9 

.11828 

.13415 

.12665 

.14502 

.13529 

.15645 

.14418 

.16848 

9 

10 

.11842 

.13-133 

.12679 

.14521 

.13543 

.15665 

.14433 

.16868 

10 

It 

.11856 

.13451 

.12694 

.14539 

.13558 

.15684 

.14449 

.16889 

11 

12 

.11870 

.13468 

.12708. 

.14558 

.13573 

.15704 

.14464 

.16909 

12 

13 

.11883 

.13486 

.12722 

.14576 

.13587 

.15724 

.14479 

.16930 

13 

14 

.11897 

.13504 

.12736 

.14595 

.13602 

.15743 

.14494 

.16950 

14 

15 

.11911 

.13521 

.12750 

.14014 

.13616 

.15763 

.14509 

.16971 

15 

16 

.11925 

.13539 

.12765 

.14632 

.13631 

.15782 

.14524 

.16992 

16 

17 

.11938 

.13557 

.12779 

.14651 

.13646 

.15802 

.14539 

.17012 

17 

18 

.11952 

.13575 

.12793 

.14670 

.13600 

.15822 

.14554 

.17033 

18 

19 

.11966 

.13593 

.12807 

.14689 

.13675 

.15841 

.14569 

.17054 

19 

20 

.11980 

.13610  i 

.12822 

.14707  i 

.13690 

.15861 

.14584 

.17075 

20 

2t 

.1*394 

.13628 

.12836 

.14726 

.13705 

.15881 

.14599 

.17095 

21 

22 

.12007 

.13046  j 

.12850 

.14745 

.13719 

.15901 

.14615 

.17116 

22 

23 

.12021 

.13664 

.12864 

.14764 

.13734 

.15920 

.14630 

.17137 

23 

24 

.12035 

.13682  j 

.12879 

.14782 

.13749 

.15940 

.14645 

.17158 

24 

25 

.12049 

.13700  ! 

.12893 

.14801 

.13763 

.15960 

.14600 

.17178 

25 

20 

.12063 

.13718  ! 

.12907 

.14820 

.13778 

.15980 

.14675 

.17199 

26 

27 

.12077 

.137:35  i 

.12921 

.14839 

.13793 

.16000 

.14090 

.17220 

27 

28 

.12091 

.13753  i 

.12936 

.14858 

.13808 

.16019 

.14706 

.17241 

28 

29 

.12101 

.13771  1 

.12950 

.14877 

.13822 

.16039 

.14721 

.17262 

29 

30 

.12118 

.13789 

.12904 

.14896 

.13837 

.16059 

.14726 

.17283 

30 

31 

.12132 

.13807 

.12979 

.14914 

.13852 

.16079 

.14751 

.17304 

31 

32 

.12146 

.13825  j 

.12993 

.14933  ! 

.13867 

.16099 

.14766 

.17325 

32 

33 

.12160 

.13843 

.13007 

.14952  ! 

.13881 

.16119 

.14782 

.17346 

33 

34 

.12174 

.13861  ) 

.13022 

.  14971 

.13896 

.16139 

.14797 

.17367 

34 

35 

.12188 

.13879  j 

.13030 

.14990 

.13911 

.16159 

.14812 

.17388 

35 

36 

.12202 

.13897 

.13051 

.15003 

.13926 

.10179 

.14827 

.17409 

36 

37 

.12216 

.13916 

.13005 

.15028 

.13941 

.16199 

.14843 

.17430 

37 

38 

.12230 

.13934 

.13079 

.15047 

.13955 

.16219 

.14858 

.17451 

38 

39 

.12244 

.13952 

.13094 

.15006 

.13970 

.16239 

.14873 

.17472 

39 

40 

.12257 

.13970 

.13108 

.15085 

.13985 

.16259 

.14888 

.17493 

40 

41 

.12271 

.13988 

.13122 

.15105 

.14000 

.16279 

.14904 

.17514 

41 

42 

.12285 

.14006 

.13137 

.15124 

.14015 

.16299 

.14919 

.17535 

42 

48 

.12299 

.14024 

.13151 

.15143 

.14030 

.16319 

.14034 

.17556 

43 

44 

.12313 

.14042 

.13106 

.15102 

.14044 

.16339 

.14949 

.17577 

44 

45 

.12327 

.14061 

.13180 

.15181 

.14059 

.16359 

.14965 

.17598 

45 

46 

.12341 

.14079 

.13195 

.15200 

.14074 

.10380 

.14980 

.17620 

40 

47 

.12355 

.14097 

.13209 

.15219 

.14089 

.10400 

.14995 

.17641 

47 

48 

.12369 

.14115 

.13223 

.15239 

.14104 

.16420 

.15011 

.17662 

48 

49 

.12383 

.14134 

.13238 

.15258 

.14119 

.10440 

.15026 

.17683 

40 

50 

.12397 

.14152 

.13252 

.15277 

.14134 

.16460 

.15041 

.17704 

50 

51 

.12411 

.14170 

.13267 

.15296 

.14149 

.16481 

.15057 

.17726 

51 

52 

.12425 

.14188 

.13281 

.15315 

.14104 

.16501 

.15072 

.17747 

52 

63 

.12439 

.14207 

.13296 

.15335 

.14179 

.16521 

.15087 

.17768 

53 

54 

.12454 

.14225 

.13310 

.15354 

.14194 

.16541 

.15103 

.17790 

54 

55 

.124C8 

.14243 

.13325 

.15373 

.14208 

.16562 

.15118 

.17811 

55 

56 

.12482 

.14262 

.13339 

.15393 

.14223 

.16582 

.15134 

.17832 

56 

57 

.12496 

.14280 

.  13354 

.15412 

.14238 

.16602 

.15149 

.17854 

57 

58 

.12510 

.14299 

.13368 

.15431 

.14253 

.16623 

.15164 

.17875 

58 

59 

.12524 

.14317 

.13383 

.15451 

.14208 

.10043 

.15180 

.17896 

59 

60 

.12538 

.14335 

.13397 

.15470 

.14283 

.10663 

.15195 

.17918 

•60 

477. 


TABLE  XXIX.— NATURAL  VERSED   SINES  AND   EXTERNAL  SECANTS. 


/ 

3 

2° 

3 

3° 

!  s 

4° 

!  * 

5° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.15195 

.17918 

.161*3 

.19230 

.17096 

.20022 

.18085 

.22077 

0 

1 

.15211 

.17939 

.10149 

.19259 

.17113 

.20045 

.18101 

.22102 

1 

2 

.15220 

.17901 

.10105 

.19281 

.17129 

.20009 

.18118 

.2:2127 

2 

3 

.15241 

.17982 

.10181 

.19304 

.17145 

.20093 

.18135 

.22152 

3 

4 

.15257 

.18004 

.16196 

.19327 

.17101 

.20717 

.18152 

.22177 

4 

5 

.15272 

.18025 

.16212 

.19349 

'.  17178 

.20740 

.18168 

.22202 

5 

G 

.15288 

.18047 

.16228 

.19372 

.17194 

.20764 

.18185 

.22227 

6 

7 

.15303 

.18008 

.16244 

.19394 

.17210 

.20788 

.18202 

.22252 

7 

8 

.15319 

.18090 

.16200 

.19417 

.17227 

.20812 

.18218 

.22277 

8 

9 

.15334 

.18111 

.16276 

.19440 

.17243 

.20836 

.18235 

.22302 

9 

10 

.15350 

.18133 

.16292 

.19463 

.17259 

.20859 

.18252 

.22327 

10 

11 

.15365 

.18155 

.16308 

.19485 

.17276 

.20883 

.18269 

.22352 

11 

12 

.15381 

.18176 

.16324 

.19508 

.17292 

.20907 

.183S6 

.28377 

13 

13 

.15396 

.18198 

.16340 

.19531 

.17808 

.20931 

.18302 

.22402 

13 

14 

.15412 

.18220 

.16355 

.19554 

.17325 

.20955 

.18319 

.22428 

14 

15 

.15427 

.18241 

.16371 

.19576 

.17341 

.20979 

.18336 

.22453 

15 

16 

.15443 

.18263 

.16387 

.19599 

.17357 

.21003 

.18353 

.22478 

16 

17 

.15458 

.18285 

.16403 

.19622 

.17374 

.21027 

.ia369 

.22503 

17 

18 

.15474 

.18307 

.16419 

.19645 

:  .17390 

.21051 

!  .18386 

.225-28 

18 

19 

.15489 

.18328 

.16435 

.19668 

.17407 

.21075 

j  .18403 

.22554 

19 

20 

.15505 

.18350 

.16451 

.19691 

:  .17423 

.21099 

1  .18420 

.22579 

20 

21 

.15520 

.18372 

.16467 

.19713 

.17439 

.21123 

.18437 

.22604 

21 

22 

.  1653(5 

.18394 

.16483 

.19736 

.17456 

.21147 

!  .18454 

.22029 

22 

23 

.15552 

.18416 

.16499 

.19759 

.17472 

.21171 

.18470 

.22055 

23 

24 

.15567 

.18437 

.16515 

.19782 

.17489 

.21195 

.18487 

.22680 

24 

25 

.15583 

.18459 

.16531 

.19805 

.17505 

.21220 

.18504 

.22706 

25 

26 

.15598 

.18481 

.16547 

.19828 

!  .17522 

.21244 

.18521 

.237-31 

26 

27 

.15614 

.1S503 

.16563 

.19851 

.17538 

.01268 

.m538 

.22756 

27 

28 

.15630 

.18525 

.16579 

.19874 

.17554 

.21292 

.18555 

.22782 

28 

29 

.15645 

.18547 

.16595 

.19897 

.17571 

.21310 

.18572 

.22F07 

29 

30 

.15661 

.18569 

.16611 

.19920 

.17587 

.21341 

.18588 

.22833 

30 

31 

.15676 

.18591 

.16627 

.19944 

!  .17604 

.21365 

.18005 

.22858 

31 

32 

.15(392 

.18613 

.16644 

.19907 

.17020 

.213S9 

.18022 

.22S84 

32 

33 

.15708 

.18635 

.16660 

.19990 

1  .17037 

.21414 

.18639 

.22009 

33 

34 

.15723 

.18657 

.16676 

.20013 

|  .17053 

.21438 

.18056 

.22935 

34 

35 

.  15739 

.18679 

.16692 

.20036 

.17670 

.21402 

.18673 

.22960 

35 

36 

.1575.") 

.18701 

.16708 

.20059 

.17086 

.21487 

.18090 

.22986 

36 

37 

.15770 

.18723 

.16724 

.20083 

.17703 

.21511 

.18707 

.23012 

37 

38 

.15780 

.18745 

,16740 

.20106 

.17719 

.21535 

.18724 

.23037 

38 

39 

.15803 

.18767 

.16756 

.20129 

.17736 

.21500 

.18741 

.23003 

39 

40 

.15818 

.18790 

.16772 

.20152 

.17752 

.31584 

.18758 

.23089 

40 

41 

.15833 

.18812 

.16788 

.20176 

.17709 

.21609 

.18775 

.23114 

41 

42 

.15849 

.18834 

.16805 

.20199 

.17786 

.21033 

.18792 

.23140 

42 

43 

.15365 

.18836 

.10821 

.20222 

.17302 

.21058 

.18809 

.23166 

43 

44 

.15880 

.18878 

.16837 

.20246 

.17819 

.21682 

.18826 

.23192 

44 

45 

.15896 

.18901 

.16853 

.20209 

.  17835 

.21707 

.18843 

.23217 

45 

46 

.15912 

.1892:) 

.16809 

.20292 

.17852 

.21731 

.18860 

.23243 

46 

47 

.15923 

.18945 

.16885 

.2C316 

.17808 

.21756 

.18877 

.23209 

47 

48 

.15943 

.18967 

.16902 

.20339 

.17885 

.21781 

.18894 

.23295 

48 

49 

.15959 

.18990  i 

.16918 

.20303 

.17902 

.21805 

.18911 

.233-21 

49 

50 

.15975 

.19012 

.16934 

.20388 

.17018 

.21830 

.18928 

.23347 

50 

51 

.15991 

.19034 

.16950 

.20110 

.17035 

.21855 

.18945 

.23373 

51 

52 

.16008 

.19057 

.16986 

.20433 

.17052 

.21879 

.18962 

.23399 

52 

53 

.10022 

.19079 

.169*3 

:  20  157 

.17908 

.21904 

.18979 

.23424 

53 

54 

.10038 

.19102 

.16909 

.20180 

.17985 

.21029 

.18996 

.23450 

54 

55 

.16054 

.19124  | 

.17015 

.20504 

.18001 

.21953 

.19013 

.23476 

55 

56 

.16070 

.19146 

.17031 

.30527 

.18018 

.21978 

.19030 

.23502 

50 

57 

.16J85 

.19169 

.17047 

.20551 

.18035 

.22003 

.19047 

.23529 

57 

58 

.16101 

.19191 

.17064 

.20575 

.18051 

.230-28 

.19004 

.23555 

58 

59 

.16117 

.19214 

.17080 

.2050S 

.18068 

.22053 

.19081 

.23581 

59 

60 

.16133 

.19330 

.17096 

.20022 

.18085 

.22077 

.19098 

.23607 

60 

478 


TABLE   XXIX.— NATURAL  VERSED   SINES   AND   EXTERNAL   SECANTS. 


3 

8° 

3' 

r° 

3 

jo 

i 

3° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.19098 

.23607 

.201S6 

.25214 

.21199 

.26902 

.22285 

.28676 

0 

1 

.19115 

.23633 

.20154 

.25241 

.21217 

.26931 

.22304 

.28706 

1 

2 

.19133 

.231)59 

.20171 

.25269 

.21235 

.26960. 

.22322 

.28737 

2 

8 

.19150 

.23685 

.20189 

.25296 

.21253 

.26988 

.22:340 

.28767 

3 

4 

.19167 

.23711 

.20207 

.25324 

.21271 

.27017 

.22359 

.28797 

4 

5 

.19184 

.23738 

.20224 

.25351 

.21289 

.27046 

.22377 

.28828 

5 

6 

.  19201 

.23704 

.20242 

.25379 

.21307 

.27075 

.22395 

.28858 

6 

.19218 

.23790 

.20259 

.25406 

.21324 

.27104 

.22414 

.28889 

7 

8 

.19235 

.23816 

.20277  . 

.25434 

.21342 

.27133 

.22432 

.28919 

8 

o 

.19252 

.23843 

.20294 

.25462 

.21360 

.27162 

.22450 

.28950 

9 

10 

.19270 

.23869 

.20312 

.25489 

.21378 

.27191 

.22469 

.28980 

10 

11 

.19287 

.23895 

.20339 

.25517 

.21396 

.27221 

.22487 

.29011 

11 

12 

.19304 

.23922 

.20347 

.25545  j 

.21414 

.27250 

.22506 

.29042 

12 

18 

.19381 

.23918 

.20305 

.25572 

.21432 

.27279 

.22524 

.29072 

13 

11 

.19338 

.23975 

.20382 

.25600 

.21450 

.27308 

.22542 

.29103 

14 

15 

.11)35(3 

.24)01 

.20400 

.25628 

.21468 

.27337 

.22561 

.29133 

15 

10 

.19373 

.24028 

.20417 

.25656 

.21486 

.27306 

.22579 

.29164 

16 

17 

.19390 

.24054 

.20435 

.25683 

.21504 

.27396 

.22598 

.29195 

17 

18 

.19407 

.24081 

.20453 

.25711 

.21522 

.27425 

.22010 

.29220 

18 

19 

.19424 

.24107 

.20470 

.25739 

.21540 

.27454 

.22634 

.29256 

19 

20 

.19442 

.24134 

.20488 

.25767 

.21558 

.27483 

.22653 

.29287 

20 

21 

.19459 

.24160 

.20506 

.25795  : 

.21576 

.27513 

.22671 

.29318 

21 

22 

.19476 

.24187 

.20523 

.25823  ! 

.21595 

.27542 

.22090 

.29319 

22 

23 

.19493 

.24213 

.20541 

.25851 

.21613 

.27572 

.22708 

.29380 

23 

24 

.19511 

.21210 

.20559 

.25879 

.21631 

.27001 

.22727 

.29411 

24 

25 

.19528 

.24267 

.20576 

.25907 

.21649 

.27630 

.22745 

.29442 

25 

26 

.19545 

.24293 

.20594 

.25935 

.21667 

.27660 

.22764 

.29473 

26 

°7 

.19562 

.24320 

.20612 

.25963 

.21685 

.27689 

.22782 

.29504 

27 

28 

.19580 

.24347 

.20029 

.25991 

.21703 

.27719 

.22801 

.29535 

28 

29 

.19597 

.24373 

.20647 

.26019 

.21721 

.27748 

.22819 

.29506 

29 

30 

.  19014 

.24400 

.20665 

.26047 

.21739 

.27778 

.22838 

.29597 

30 

31 

.19032 

.24427 

.20682 

.26075 

.21757 

.27807 

.22856 

.29628 

31 

32 

.19049 

.21454 

.20700 

.26104  : 

.21775 

.27837 

.22875 

.29659 

32 

33 

.19666 

.24481 

.20718 

.26132  ! 

.21794 

.27867 

.22893 

.29090 

33 

34 

.1J084 

.24508 

.20736 

.26160 

.21812 

.27896 

.22912 

.29721 

34 

35 

.19701 

.24534 

.20753 

.26188 

.21830 

.27926 

.22030 

.29752 

35 

36 

.19718 

.24561 

.20771 

.26216 

.21848 

.27956 

.22949 

.29784 

36 

37 

!  19736 

.24588 

.20789 

.26245 

.21866 

.27985 

.22967 

.29815 

37 

38 

.  19753 

.24615 

.20807 

.26273 

.21884 

.28015 

.22986 

.29846 

38 

39 

.19770 

.21642 

.20824 

.26301 

.21902 

.28045 

.23004 

.29877 

39 

40 

.19788 

.21009 

.20842 

.26330 

.21921 

.28075 

.23023 

.29909 

40 

41 

.19805 

.24696 

.20860 

.26.358 

.21939 

.28105 

.23041 

.29940 

41 

42 

.19822 

.24723 

.20878 

.26387 

.21957 

.28134 

.23060 

.29971 

42 

43 

.19840 

.24750 

.20895 

.26415 

.21975 

.28164 

.2*379 

.30003 

43 

44 

.19857 

.24777 

.20913 

.26443 

.21993 

.28194 

.23097 

.30034 

44 

45 

.19875 

.24804 

.20931 

.26472 

.22012 

.28224 

.23116 

.30066 

45 

46 

.19892 

.24832 

.20949 

.26500 

.22030 

.28254  i 

.23134 

.30097 

46 

47 

.19909 

.24859 

.20967 

.26529 

.22048 

.28284  j 

.23153 

.30129 

47 

48 

.19927 

.24886 

.20985 

.26557 

.22066 

.28314  ! 

.23172 

.30160 

48 

49 

.19944 

.24913 

.21002 

.26586 

.22084 

.28344  i 

.23190 

.30192 

49 

50 

.19962 

.24940 

.21020 

.26615 

.22103 

.28374  ' 

.23209 

.30223 

50 

51 

.19979 

.24967 

.21038 

.26643 

.22121 

.28404 

.23228 

.30255 

51 

52 

.19997 

.24995 

.21056 

.20072 

.22139 

.28434 

.23246 

.30287 

62 

53 

.20014 

.25022 

.21074 

.26701 

.22157 

.28464  : 

.23265 

.30318 

53 

54 

.20032 

.25049 

.21092 

.26729 

.22176 

.28495 

.23283 

.30350 

54 

55 

.20049 

.25077 

.21109 

.26758  : 

.22194 

.28525  i 

.23302 

.30382 

55 

56 

.20066 

.25104 

.21127 

.20787  ! 

22212 

.28555 

.23321 

.30413 

56 

57 

.20084 

.25131 

.21145 

.26815  1 

!  22231 

.28585  i 

.23339 

.30445 

57 

58 

.20101 

.25159 

.21163 

.26844  i 

.22249 

.28615 

.2:3358 

.30477 

58 

59 

.20119 

.25186  i 

.21181 

.26873 

.22207 

.28646 

.23377 

.30509 

59 

60 

.20136 

.25214 

.21199 

.20902  | 

.22285 

.28076 

.23396 

.30541 

€0 

479 


TABLE  XXIX.—  NATURAL  VERSED  SINES  AND   EXTERNAL   SECANTS 


/ 

40° 

«• 

42° 

43° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers.  Ex.  sec. 

Vers. 

Ex.  sec. 

~0 

.23396 

.30541 

.2-1529  1  .32501 

.25686 

.34563 

.26865 

.36733 

0 

1 

.23414 

.30573 

.24548   .32535 

.25705 

.34599 

.26884 

.36770 

1 

2 

.23433 

.30605 

-.24567 

.32568 

.257'24 

.34634 

.26904 

.36807 

2 

3 

.23452 

.30036 

.24586 

.32C02 

.25744 

.34669 

.26924 

.36844 

3 

4 

.23470 

.30668 

.24605 

.32636 

.25763 

.34704 

.26944 

.36881 

4 

5 

.23489 

.30700 

.24625 

.32669 

.25783 

.34740 

.26964 

.36919 

5 

6 

.23508 

.30732 

.24644 

.32703  j 

.25802 

.34775 

.26984 

.36956 

6 

7 

.23527 

.30764 

.24663 

.32737 

.25822 

.34811 

.27004 

.36993 

7 

8 

.23545 

.30796 

.24682 

.32770 

.25841 

.34846 

.27084 

.37030 

8 

9 

.23564 

.30829 

.24701 

.32804 

.25861 

.34882 

.27043 

.37068 

9 

10 

.28583 

.30861 

.24720 

.32838 

.25880 

.34917 

.27063 

.37105 

10 

11 

.23602 

.30893 

.24739 

.32872 

.25900 

.24353 

.27083 

.37143 

11 

12 

.23620 

.30925 

.247'59 

.32905 

.25920 

.34988 

.27103 

.37180 

12 

13 

.23639 

.30957 

.24778 

.32939 

.25989 

.35024 

.27123 

.37218 

13 

14 

.23658 

.30989 

.24797 

.32973 

.25959 

.35060 

.27143 

,37'255 

14 

15 

.23677 

.31022 

.24816 

.33007 

.25978 

.35095 

.27163 

.87293 

15 

16 

.23696 

.31054 

.24835 

.33041 

.25998 

.35131 

.27183 

.37330 

16 

17 

.23714 

.31086 

.24854 

.33075 

.26017 

.35167 

.27'203 

.37368 

17 

18 

.23733 

.31119 

.24874 

.33109 

.26037 

.35203 

.27223 

.37406 

18 

19 

.23752 

.31151 

.24893 

.33143 

.26056 

.35238 

.27243 

.37443 

19 

98 

.23771 

.31183 

.24912 

.33177 

.26076 

.35274 

.27263 

.37481 

20 

21 

.23790 

.31216 

.24931 

.33211 

.26096 

.35310 

.27283 

.37519 

21 

22 

.23808 

.31248 

.24950 

.33245 

.26115 

.35346 

.27303 

.37556 

22 

23 

.23827 

.31281 

.24970 

.33279 

.26135 

.35382 

.27823 

.37594 

23 

24 

.23346 

.31313 

.24989 

.a3314 

.26154 

.35418 

.27343 

.37632 

24 

25 

.23865 

.31346 

.25008 

.33348 

.26174 

.35454 

.27363 

.37670 

25 

26 

.23884 

.31378 

.25027 

.33382 

.26194 

.35490 

.27383 

.37708 

26 

27 

.23903 

.31411 

.25047 

.83416 

.26213 

.35526 

.27403 

.37746 

27 

28 

.23922 

.31443 

.25066 

.33451 

.26233 

.353(52 

.27423 

.87784 

28 

29 

.23941 

.31476 

.25085 

.33485 

.26253 

.35598 

.27443 

.37822 

29 

30 

.23959 

.31509 

.25104 

.33519 

.26272 

.35634 

.27463 

.37860 

30 

31 

.23978 

.31541 

.25124 

.33554 

.26292 

.35670 

.27483 

.37898 

31 

32 

.23997 

.31574 

.25143 

.33588 

.26312 

.35707 

.27503 

.37936 

32 

33 

.24016 

.31607 

.25162 

.33622 

.26331 

.35743 

.27528 

.37974 

33 

34 

.24083 

.31610 

.25182 

.33657 

.26351 

.35779 

.27543 

.38012 

34 

35 

.24054 

.31672 

.25201 

.33691 

.26371 

.35815 

.27563 

.38051 

35 

36 

.24073 

.31705 

.25220 

.33726 

.26390 

.35H52 

.27583 

.38089 

36 

37 

.24092 

.31738 

.25240 

.-33760 

.26410 

.35888 

.27603 

.38127 

37 

38 

.24111 

.31771 

.25259 

.33795 

.26430 

.35924 

.27'623 

.38165 

38 

39 

.24130 

.31804 

.25278 

.33830 

.26449 

.35961 

.27043 

.38204 

39 

40 

.24149 

.31837 

.25297 

.33864 

.26469 

.35997 

.27063 

.38242 

40 

41 

.24168 

.3187'0 

.25317 

.3:5899 

.26489 

.36034 

.27683 

.38280 

41 

42 

.24187 

.31903 

.25336 

.33934 

.26509 

.36070 

.277-03 

.38319 

42 

43 

.24206 

.31936 

.25356 

.33968 

.26528 

.36107 

.27723 

.38357 

43 

44 

.24225 

.31969 

.25375 

.34003 

.26548 

.36143 

.27743 

.38396 

44 

45 

.24244 

.32002 

.25394 

.34038 

.26568 

.36180 

.27764 

.38434 

45 

46 

.24262 

.32035 

.25414 

.34073 

.26588 

.36217 

.27784 

.3&47'3 

46 

47 

.24281 

.32068 

.25433 

.34108 

.26607 

.36253 

.27804 

.38512 

47 

48 

.2-1300 

.32101 

.25452 

.34142 

.20627 

.36290 

.27824 

.38550 

48 

49 

.24320 

.32134 

.25472 

.34177 

.26647 

.36327 

.27844 

.38589 

49 

50 

.24339 

.32168 

.25491 

.34212 

.26667 

.36363 

.27804 

.38028 

50 

51 

.24358 

.32201 

.25511 

.34247 

.26686 

.36400 

.27884 

.38600 

51 

52 

.21377 

.32234 

.25530 

.34282 

.26706 

.36437 

.27905 

.88705 

52 

53 

.24396 

.,32267 

.25549 

.34317 

.26726 

.36474 

.27925 

.38744 

53 

54 

.21415 

.32301 

.25569 

.34352 

.21)746 

.36511 

.27945 

.38783 

54 

55 

.21134 

.32334 

.25588 

.34387 

.2(5766 

.36548 

.27'965 

.38822 

55 

56 

.24453 

.32368 

.25608 

.34423 

.26785 

.36585 

.27985 

.38860 

56 

57 

.24472 

.32401 

.25627 

.34458 

.26805 

.36622 

.28005 

.38899 

57 

58 

.24491 

.32434 

.25647 

.34493 

.26825 

.36659 

.28026 

.83938 

58 

59 

.24510 

.32468 

.25666 

.34528 

.26845 

.30696 

..28046 

.38977 

59 

60 

.24529 

.32501 

.25686 

.34563 

.26805 

.36733 

.28006 

.39016 

60 

480 


TABLE    XXIX. -NATURAL  VERSED   SINES  AND  EXTERNAL   SECANTS. 


4 

4° 

4 

5° 

4 

6° 

4 

7° 

Yers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Yers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.28066 

.39016 

1  .29289 

.41421 

.305:34 

.43956 

.31800 

.46628 

0 

1 

.23086 

.39055 

.29310 

.41463 

.30555 

.43999 

.31821 

.46674 

1 

2 

.28106 

.39095 

i  .29330 

.41504 

.30576 

.44042 

.31843 

.46719 

2 

3 

.28127 

.39134 

i  .29351 

.'41545 

.30597 

.44086 

.31864 

.46765 

3 

4 

.28147 

.39173 

i  .29372 

.41586 

.30618 

.44129 

.31885 

.46811 

4 

5 

.23167 

.39212 

1  .29392 

.41627 

.30639 

.44173 

.31907 

.46857 

5 

6 

.28187 

.39251 

i  .29413 

.41669 

.30660 

.44217 

.31928 

.46903 

6 

7 

.28208 

.39291 

.29433 

.41710 

.30681 

.44260 

.31949 

.46949 

7 

8 

.28228 

.39330 

.29454 

.41752 

.30702 

.44304 

.31971 

.46995 

8 

9 

.28248 

.39369 

.29475 

.41793 

.30723 

.44347 

.31992 

.47041 

9 

10 

.23268 

.39409 

.29495 

.41835 

.30744 

.44391 

.32013 

.47087 

10 

11 

.28289 

.39448 

.29516 

.41876 

.30765 

.44435 

.32035 

.47134 

11 

12 

.28:309 

.39487 

.29537 

.41918 

.30786 

.44479 

.32056 

.47180 

12 

13 

.28329 

.39527 

.29557 

.41959 

.30807 

.44523 

.32077 

.47226 

13 

14 

.28350 

.39566 

.29578 

.42001 

.30828 

.44567 

.32099 

.47272 

14 

15 

.28370 

.39806 

.29599 

.42043 

.30849 

.44610 

.32120 

.47319 

15 

16 

.28390 

.39646 

.29619 

.42084 

.30870 

.44654 

.32141 

.47365 

16 

17 

.28410 

.39685 

.29640 

.42126 

.  .30891 

.44698 

.32163 

.47411 

17 

18 

.28431 

!  39725 

.29661 

.42168 

.30912 

.44742 

.32184 

.47458 

18 

10 

.23451 

.39764 

.29881 

.42210 

.30933 

.44787 

.32205 

.47504 

19 

20 

.28471 

.39804 

.29702 

.42251 

.30954 

.44831 

.32227 

.47551 

20 

21 

.28492 

.39844 

.29723 

.42293 

.30975 

.44875 

.32248 

.47598 

21 

22 

.28512 

.89884 

.29743 

.42*35 

.30996 

.44919 

.32270 

.47644 

22 

23 

.2^52 

.39924 

.29764 

.42377 

.31017 

.44963 

.32291 

.47691 

23 

24 

.28553 

.39963 

.29785 

.42419 

.31038 

.45007 

.32312 

.47738 

24 

25 

.  28573 

.40003 

.29805 

.42461 

.31059 

.45052 

.32334 

.47784 

25 

26 

.28593 

.40043 

.29826 

.42503 

.31080 

.45096 

.32355 

.47831 

26 

27 

.28614 

.40083 

.29847 

.42545 

.31101 

.45141 

.32377 

.47878 

27 

28 

.28634 

.40123 

.29868 

.42587 

.31122 

.45185 

.32398 

.47925 

28 

29 

.28655 

.40163 

.29888 

.42630 

.31143 

.45229 

.32420 

.47972 

29 

30 

.28675 

.40203 

.29909 

.42672 

.31165 

.45274 

.32441 

.48019 

30 

31 

.28695 

.40243 

.29930 

.42714 

.31186 

.45319 

.32462 

.48066 

31 

32 

.28716 

.40283 

.29951 

.42756 

.31207 

.45363 

.32484 

.48113 

32 

33 

.28736 

.40324 

.29971 

.42799 

.31228 

.45408 

.32505 

.48160 

33 

34 

.23757 

.40364 

.29992 

.42841 

.31249 

.45452 

.32527 

.48207 

34 

35 

.28777 

.40404 

.30013 

.42883 

.31270 

.45497 

.32548 

.48254 

35 

36 

.28797 

.40444 

.30034 

.42926 

.31291 

.45542 

.32570 

.48301 

36 

37 

.28818 

.40485 

.30054 

.42968 

.31312 

.45587 

.32591 

.48349 

37 

38 

.28838 

.40525 

.30075 

.43011 

.31334 

.45631 

.32613 

.48396 

38 

39 

.28859 

.40565 

.30096 

.43053 

.31355 

.45676 

.3263-4 

.48443 

39 

40 

.28879 

.40606 

.30117 

.43096 

.31376 

.45721 

.32656 

.48491 

40 

41 

.28900 

.40646 

.30138 

.43139 

.31397 

.45766 

.32677 

.48538 

41 

42 

.28920 

.40687 

.30158 

.43181 

.31418 

.45811 

.32699 

.48586 

42 

43 

.28941 

.40727 

.30179 

.43224 

.31439 

.45856 

.32720 

.48633 

43 

44 

.28961 

.40768 

.30200 

.43267 

.31461 

.45901 

.32742 

.48681 

44 

45 

.23981 

.40808 

30221 

.43310 

.31482 

.45946 

.32763 

.48728 

45 

46 

.29002 

.40849 

.'30242 

.43352 

.31503 

.45992 

.32785 

.48776 

46 

47 

.29023 

.40890 

.30263 

.43395 

.31524 

.46037 

.32806 

.48824 

47 

48 

.29043  H 

.40930 

.30283 

.43438 

.31545 

.46082 

.32828 

.48871 

48 

49 

.29063 

.40971 

.30304 

.43481 

.31567 

.46127 

.32849 

.48919 

49 

50 

.29034 

.41012 

.30325 

.43524 

.31588 

.46173 

.32871 

.48967 

50 

51 

.29104 

.41053 

.30346 

.43567 

.31609 

.46218 

.32893 

.49015 

51 

52 

.29125 

.41093 

.30367 

.43610 

.31630 

.46263 

.32914 

.49063 

52 

53 

.29145 

.41134 

.30388 

.43653  i 

.31651 

.46309 

.32936 

.49111 

53 

54 

,29166 

.41175 

.30409 

.43696  ! 

.31673 

.46354 

.32957 

.49159 

54 

55 

.29187 

.41216 

.30430 

.43739,  ; 

.31694 

.46400 

.32979 

.49207 

55 

56 

.29207 

.41257 

.30451 

.43783  i 

.31715 

.46445 

.33001 

.49255 

56 

57 

.29228 

.41298 

.30471 

.43826 

.31736 

.46491 

.33022 

.49303 

57 

53 

.21)248 

.41339 

.30492 

.43869 

.31758 

.46537 

.33044 

.49351 

58 

59 

.29269 

.41380 

.30513 

.43912 

.31779 

.46582 

.33065 

.49399 

59 

69 

.29289 

.41421  ! 

.30534 

.43956 

.31809 

.46628  1 

.33987 

.49448 

60 

481 


TABLE  XXIX. -NATURAL  VERSED    SINES    AND  EXTERNAL   SECANTS. 


48°         49° 

50° 

51° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

|  Vers. 

Ex.  sec. 

0 

.33087 

.49448  i  .34394 

.52425 

.35721 

.55572 

.37068 

.58902 

~0~ 

1 

.33109 

.49496 

.34416 

.52476 

.35744 

.55626 

.37091 

.58959 

1 

2 

.33130 

.49544 

.34438 

.52527 

.85766 

.55680 

.37113 

.59016 

2 

3 

.33152 

.49593 

.34460 

.38579 

.85788 

.55734 

.37136 

.59073 

3 

4 

.33173 

.49641 

.34482 

.52630 

.35810 

.55789 

.:•',;•  ir>s 

.59130 

4 

5 

.33195 

.49690 

.34504 

.52681 

.35833 

.55843 

.37181 

.59188 

5 

6 

!  83217 

.49738 

.34526 

.52732 

.35855 

.55897 

.37204 

.5<)215 

6 

7 

.83288 

.49787 

.34548 

.52781 

.35877 

.55951 

.87226 

.59302 

7 

8 

.33260 

.49835 

.34570 

.52835 

.35900 

.56005 

.37249 

.59360 

8 

9 

.a3282 

.49884 

.34592 

.52886 

.35922 

.56060 

.37272 

.59418 

9 

10 

.33303 

.49933 

.34614 

.52938 

.35944 

.56114 

.37294 

.59475 

10 

11 

.88386 

.49981 

.34636 

.52989 

.35967 

.56169 

.37317 

.59533 

11 

12 

.33347 

.50030 

.34658 

.53041 

.35989 

.56223 

.37340 

.59590 

12 

13 

.33368 

.50079 

.34680 

.53092 

.36011 

.56278 

.37362 

.59618 

13 

14 

.33390 

.50128 

.34702 

.53144 

.36034 

.56332 

.37385 

.59706 

14 

15 

.33412 

.50177 

.34724 

.53196 

.36056 

.56387- 

.37408 

.59764 

15 

1(5 

.33434 

.50226 

.34746 

.53247 

.36078 

.58442 

.37130 

.59822 

16 

ir 

.33455 

.50275 

.34768 

.53299 

.36101 

.56497 

.37453 

.59880 

17 

18 

.33477 

.50324 

.34790 

.53351' 

.36123 

.56551 

.37476 

.59938 

18 

19 

.33199 

.50373 

.34812 

.53403 

.86146 

.50606 

.37498 

.59996 

19 

20 

.33520 

.50422 

.34834 

.53455 

.3016? 

.56661 

.37521 

.60054 

20 

21 

.33542 

.50471 

.34856 

.53507 

.36190 

.56716 

.37544 

.00112 

21 

22 

.33564 

.50521 

.34878 

.53559 

.30-213 

.56771 

.37567 

.00171 

22 

23 

.33586 

.50570 

.34900 

.53611 

.36235 

.56826 

.37589 

.60229 

23 

24 

.33607 

.50619 

.34923 

.53663 

.36258 

.50881 

.37612 

.60287 

24 

25 

.33629 

.50669 

.34945 

.53715 

.36280 

.56937 

.376:35 

.60346 

25 

26 

.33651 

.50718 

.34967 

.53768 

.36302 

.56992 

.37658 

.00404 

26 

O1"* 

.33673 

.50767 

.34989 

.53820 

.36325 

.57047 

.37080 

.60-463 

27 

28 

.33694 

.50317 

.35011 

.53872 

.36347 

.57103 

.37703 

.(50521 

28 

29 

.33716 

.50866 

.35033 

.53924 

.36370 

.57158 

.37726 

.60580 

29 

30 

.33738 

.50916 

.35055 

.53977 

.36392 

.57213 

.37749 

.60639 

30 

31 

.33760 

.50966 

.35077 

.54029 

.36415 

.57269 

.37771 

.60698 

31 

32   .33782 

.51015 

.35099 

.54082 

.36437 

.57324 

.37794 

.00756 

32 

33   .33803 

.51065 

.35122 

.54134 

.36460 

.57380 

.37817 

.60815 

33 

31   .33325 

.51115 

.35144 

.54187 

.36482 

.57436 

.37840 

.60874 

34 

35   .33847 

.51165 

.35166 

.54240 

.36504 

.57491 

.37802 

.60933 

35 

36  1  .33869 

.51215 

.35188 

.54292 

.36527 

.57547 

.37885 

.60992 

36 

37 

.33891 

.51265 

.35210 

.54345 

.36549 

.57603 

.37908 

.61051 

37 

38 

.33912 

.51314 

.35232 

.54398 

.86573 

.57659 

.37931 

.61111 

38 

39 

.33934 

.51364 

.35254 

.54451 

.36594 

.57715 

.37954 

.61170 

39 

40 

.33956 

.51415 

.35277 

.5450i 

.36617 

.57771 

.37976 

.61229 

40 

41   .33978 

.51465 

.35299 

.54557 

.36639 

.57827 

.37999 

.61288 

41 

42 

.34000 

.51515 

.35321 

.54610 

.36662 

.57883 

.38022 

.61348 

42 

43 

.34022 

.51565 

.35343 

.54663 

.36084 

.57939 

.38045 

.01407 

43 

44 

.34044 

.51615 

.35365 

.54716 

.36707 

.57995 

.38068 

;  61467 

44 

45 

.34065 

.51665 

.35388 

.54769 

.36729 

.5.8051   .3S091 

.61526 

45 

46 

.34087 

.51716 

.35410 

.54822 

.36752 

.58108  :  .38113 

.61586 

46 

47 

.34109 

.51766 

.35432 

.54876 

.36775 

.58164 

.38136 

.61646 

47 

48 

.34131 

.51817 

.35454 

.54929 

.36797 

.58221 

.38159 

.61705 

48 

49 

.31153 

.51867 

.35476 

.5-19S2 

.36820 

.58277  i 

.38182 

.61705 

49 

50 

.34175 

.51918 

.35499 

.55036 

.36842 

.58333  i 

.38205 

.61825 

50 

51 

.34197 

.51968 

.35521 

.55089  !  .368C5 

.58390 

.88228 

.61885 

51 

52 

.34219 

.52019 

.35543 

.55143 

;  .36887 

.58-147 

.3S251 

.61945 

52 

53 

.34241 

.52069 

.35565 

.55196 

.36910 

.58503 

.3827'4 

.62005 

53 

54 

.34202 

.52120 

.awSS 

.55250 

.36932 

.58560 

.38296 

.62005 

54 

55 

.34284 

.52171 

.35610 

.55303 

.36955 

.58617 

.38319 

.02125 

55 

56 

.34306 

.52222 

.35632 

.55357 

.36978 

.58674 

.38342 

.62185 

56 

57 

.34328 

.52273 

.35654 

.55411 

.37000 

.58731 

.38365 

.62046 

57 

58 

.34350 

.52323 

.35677 

.55465 

.37023 

.58788 

.38388 

.62306 

58 

59 

.34372 

.52374 

.35699 

.55518 

.37045 

.58845 

.38411 

.62?,GO 

59 

CO 

.34394 

.52425 

.35721 

.55372 

.37068 

.58902 

.38434 

.62427 

CO 

4S2 


TABLE  XXIX.— NATURAL    VERSED    SINES   AND   EXTERNAL   SECANTS. 


5 

»• 

, 
5 

3° 

5' 

1° 

5 

)° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

1  Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

i 



0 

.38434 

.02427 

.39819 

.00104 

!  .41221 

.70130 

42642 

.74345 

0 

1 

.38457 

.62487 

.39842 

166228 

<  .41245 

.70198 

.42666 

.74417 

1 

o 

.38480 

.  62548 

.39865 

.66292 

.41209 

.70267 

.42690 

.74490 

2 

3 

138508 

.62609 

.39888 

.66357 

i  .41292 

.70335 

.42714 

.74562 

3 

4 

.38526 

.62669 

.39911 

.66421 

j  .41316 

.70403 

.42738- 

.74635 

4 

5 

.38649 

.62730 

.39935 

.66486 

i  .41339 

.70472 

.42702 

.74708 

5 

6 

.38571 

.62791 

.39958 

.66550 

.41363 

.70540 

.42785 

.74781 

6 

7 

.38594 

.62852 

.39981 

.66615 

.41386 

.70009 

.42809 

.74854 

7 

8 

.38017 

.62913 

.40005 

.66679 

.41410 

.70677 

i  .42833 

.74927 

8 

9 

.38640 

.62974 

.40028 

.66744 

.41433 

.70746 

1  .42857 

.75000 

9 

10 

.38663 

.63035 

.40051 

.66809 

.41457 

.70815 

.42881 

.75073 

10 

11 

.38086 

.63096 

.40074 

.66873 

.41481 

.70884 

.42905 

.75146 

11 

12 

.38709 

.63157 

.40098 

.66938 

!  .41504 

.70953 

.42929 

.75219 

12  ' 

13 

.88732 

.63218 

.40121 

.67003 

'  .41528 

.71022 

.42953 

.75293 

13 

14 

.38755 

.03279 

.40144 

.67068 

.41551 

.71091 

.42976 

.75366 

14 

15 

.38778 

.63811 

.40168 

.67133 

.41575 

.71160 

.43000 

.75440 

15 

16 

.38801 

.63402 

.401  91 

.67199 

.41599 

.71229 

.43024 

.75513 

16 

17 

.38824 

.63164 

.40214 

.67264 

.41622 

.71298 

.43048 

.75587 

17 

18 

.38847 

.63525 

.40237 

.67329 

.41646 

.71368 

.43072 

.75661 

18 

19 

.38870 

.63537 

.40201 

.67394 

.41670 

.71437 

.43096 

.75734 

19 

20 

.38893 

.63648 

.40284 

.67460 

i  .41693 

.71506 

.43120 

,75808 

20 

21 

.38910 

.63710 

.40307 

.67525 

.41717 

.71576 

.43144 

.75882 

21 

22 

.38939 

.63772 

.40331 

.67591 

.41740 

.71646 

'.43168 

.75956 

22 

23 

.38962 

.63834 

.40354 

.67656 

.41764 

.71715 

.43192 

.76031 

23 

24 

.38985 

.63895 

.40378 

.67722 

.41788 

.71785 

.43216 

.76105 

24 

25 

.39009 

.63957 

.40401 

.67788 

.41811 

.71855 

.43240 

.76179 

25 

26 

.39032 

.64019  j 

.40424 

.67853 

i  .4iass 

.71925 

.43264 

.76253 

26 

27 

.39055 

.64081 

.40448 

.67919 

.41859 

.71995 

.43287 

.76328 

27 

28 

.39078 

.04144 

.40471 

.67985 

.41882 

.72065 

.43311 

.76402 

28 

29 

.39101 

.04200 

.40494 

.68051 

.41906 

.72135 

.43335 

.76477 

29 

30 

.39124 

.64268  I 

.40518 

.68117 

.41930 

.72205 

.43359 

.76552 

30 

31 

.39147 

.64330 

.40541 

.68183 

.41953 

.72275 

.43383 

.76626 

31 

32 

.39170 

.01:593 

.40565 

.68250 

.41977 

.72346 

.43407 

.76701 

32 

33 

.39193 

.64455 

.40588 

.68316 

.42001 

.73416 

.43431 

.76776 

33 

34 

.39216 

.64518 

.40011 

.68382 

.42024 

.72487 

.43455 

.76851 

34 

85 

.39239 

.64580 

.40635 

.68449 

.42048 

.72557 

.43479 

.76926 

35 

•36 

.39202 

.64643 

.40658 

.68515 

1  .42072 

.72628 

.43503 

.77001 

36 

37 

.39286 

.64705 

.40682 

.68582 

.42096 

.72698 

.43527 

.77077 

37 

38 

.39309 

.04708 

.40705 

.68648 

.42119 

.72769 

.43551 

.77152 

38 

89 

.39333 

.64831 

.40728 

.68715 

.42143 

.72840  1 

.43575 

.77227 

39 

40 

.39355 

.64894 

.40752 

.68782 

.42167 

.72911 

.43599 

.77303 

40 

41 

.39378 

.64957 

.40775 

.68848 

.42191 

.72982 

.43623 

.77378 

41 

42 

.39401 

'  .65020 

.40799 

.68915 

.42214 

.73053  j 

.43647 

.77454 

42 

4.'] 

.39124 

.65083 

.40822 

.68982 

.42238 

.73124  i 

.43671 

.77530 

43 

44 

.39447 

.65146 

.40846 

.69049 

.42262 

.73195 

.43695 

.77606 

44 

45 

.39471 

.65209 

.40869 

.69116 

.42285 

.73267  ! 

.43720 

.77681 

45 

46 

.39494 

.65272 

.40893 

.69183 

.42309 

.73338  i 

.43744 

.77757 

46 

47 

.39517 

.65336 

.40916 

.69250 

.43333 

.73409 

.43768 

.77833 

47 

48 

.39540 

.65399 

.40939 

.69318 

.42357 

.73481 

.43792 

.77910 

48 

49 

.39563 

.65402 

.40963 

.69a85 

.42381 

.73552 

.43816 

.77986 

49 

50 

.39586 

.65526 

.40986 

.69452 

.42404 

.73624 

.43840 

.78062 

50 

51 

.39010 

.65589  ! 

.41010 

.69520 

.42428 

.73696 

.43864 

.78138 

51 

5-J 

.39033 

.05653 

.41033 

.69587 

.42452 

.7.3768 

.43888 

.78215 

52 

58 

.39656 

.05717 

.41057 

.69655 

.42476 

.73840 

.43912 

.78291 

53 

54 

.39079 

.65780 

.41080 

.69723 

.42499 

.73911 

.43936 

.78368 

54 

55 

.39702 

.058-14 

.41104 

.69790 

.42523 

.73983 

.43960 

.78445 

55 

50 

.39720 

.65908 

.41127 

.69858 

.42547 

74056 

.43984 

.78521 

56 

57 

.39749 

.65972 

.41151 

.69926 

.42571 

.74128 

.44008 

.78598 

57 

58 

.39772 

.66036 

.41174 

.69994 

.42595 

.74200 

.44032 

.78675 

58 

59 

.39795 

.66100 

.41198 

.70062 

.42019 

.7427'2 

.44057 

.78752 

59 

60 

.39819 

.66164 

.41221 

.70130 

.42042 

.74345 

.44081 

.78829 

60 

483 


TABLE    XXIX. -NATURAL  VERSED    SINES   AND   EXTERNAL   SECANTS. 


' 

56° 

57° 

58° 

59° 

' 

Vers. 

Ex.  sec. 

Vers.  'Ex.  sec. 

Vers.  Ex.  sec. 

Vers.  Ex.  sec. 

0  !  .44081 

.78829 

.45536   .83608 

.47008   .88708 

.48496  .  .94160 

0 

1  I  .44105 

.78906 

.45560  i  .83690 

.47033  !  .S8796 

.48521  i  .942.54 

1 

2  !  .44129 

.78984 

.45585 

.83773 

.47057  |  .88884 

.48546  ;  .94:349 

>> 

3  I  .44153 

.79061 

.45609   .8:3855 

i  .47082   .88972 

.48571   .94443 

3 

4  i  .44177 

.79138 

.45634   .83938 

i  .47107   .89060   .48596  :  .94537 

4 

5 

.44201 

.79216 

.45658   .84020 

!  .47131  !  .89148   .48621   .94632 

5 

6 

.44225 

.79293 

.45683 

.84103 

.47156   .89237   .48640   .94726 

6 

7 

.44250   .79371 

.45707 

.84186 

.47181  :  .89325   .48671   .94821 

rt 

8 

.44274 

.79449 

.45731 

.84269 

.47206   .89414   .48696   .94916 

8 

9 

.44298 

.79527 

.45756 

.84352 

.47230   .89503  1  1  .48721  i  .95011 

9 

10 

.44322 

.79604 

.45780 

.84435 

.47255 

.89591  h  .48746   .95106 

10 

11 

.44346   .79682 

.45805 

.84518 

i  .47280   .89680  !  .48771 

.95201 

11 

12 

.44370 

.79761 

.45829  !  .84601 

.47304 

.89769  1  .48796 

.95296 

12 

13 

.44395   .79839 

.45854   .84685 

.47329 

.89858  I1  .48821 

.95392 

13 

14 

.44419   .79917 

.45878   .84768 

.47354 

.89948  i  .48846 

.95487 

14 

15 

.44443   .79995 

.45903 

.84852 

.47379 

.90037   .48871 

.95583 

15 

16 

.44467   .80074 

.45927 

.84935 

.47403   .901261  -48896 

.95678 

16 

17 

.44491   .80152 

.45951 

.85019 

1  .474.28 

.90216  I  .48921   .95774 

17 

18 

.44516 

.80231 

.45976 

.85103 

.47453 

.90305 

.48946   .95870 

18 

19 

.44540 

.80309 

.46000 

.85187 

!  .47478 

.90395 

.48971   .95966 

19 

20 

.44564   .80388 

.46025 

.85271 

i  .47502 

.90485   .48996 

.96062 

20 

21 

.44588   .80467 

.46049 

.85355 

i  .47527 

.90575   .49021 

.96158 

21 

23 

.44612 

.80546 

.46074 

.85439 

.47552 

.90665 

.49046 

.96255 

22 

23 

.44637 

.80625 

.46098 

.85533 

.47577 

.90755   .49071 

.96351 

23 

24 

.44661 

.80704 

.46123  i  .85608 

.47601 

.90845 

.49096  i  .96448 

24 

25 

.44685 

.80783 

.46147  i  .85692 

.47626 

.90935 

.49121   .96544 

25 

26 

.44709 

.80862 

.46172   .85777 

.47651 

.91026 

.49146   .96641 

26 

27 

.44734 

.80942  | 

.46196 

.85861 

.47676 

.91116  i  .49171   .96738 

27 

28 

.44758 

.81021 

.46221 

.85946 

.47701 

.91207  !  .49196   .96835 

28 

29 

.44782 

.81101 

.46246 

.b6031 

.47725 

.91297  '  .49221  i  .96932 

29 

30 

.44806 

.81180 

.46270 

.86116 

.47750 

.91388 

.49246  I  .97029 

30 

31 

.44831 

.81260 

.46295 

.86201 

.47775 

.91479 

.49271   .97127 

31 

32 

.44855 

.81340 

.46319 

.86286 

.47800 

.91570  ;  .49296 

.97224 

32 

33 

.44879 

.81419 

.46344 

.86371 

.47825 

.91(561  !  .49321 

.97'322 

33 

34 

.44903 

.81499 

.46368 

.86457 

!  .47849 

.91752  i  .49346   .97420 

34 

35 

.44923 

.81579 

.46393 

.86542 

.47874   .91844   .49372  j  .97517 

35 

36 

.44952 

.81659 

.46417 

.86627 

.47899   .91935  •  .49397   .97615 

36  • 

37 

.44976 

.81740 

.46442 

.86713 

!  .47924   .92027   .49422  j  .97713 

37* 

38 

.45001 

.81820 

.46466 

.86799 

1  .47949 

.92118  i  .49447   .97811 

38 

39 

.45025 

.81900 

.46491 

.86885 

.47974 

.92210  l!  .4947'2  i  .97910 

39 

40 

.45049 

.81981 

.46516 

.86990 

.47998 

.92302 

i  .49497   .98008 

40 

41 

.45073 

.82061 

.46540   .87056 

1  .48023 

.92394 

1  .49522  !  .98107 

41 

42 

.45098 

.82142 

.46565   .87142 

i  .48048   .92486 

i  .49547   .98205 

42 

43 

.45122 

.82222 

.46589 

.87229 

1  .48073   .92578 

.49572   .9830-1 

43 

44 

.45146 

.82303 

.46614 

.87315 

:  .48098 

.92670 

.49597 

.98403 

44 

45 

.45171 

.82384 

.46639 

.87401 

.48123 

.92762 

.49623 

.98502 

45 

46 

.45195 

.82465 

.46663 

.87488 

i  .48148 

.92855 

i  .49648   .98601 

46 

47 

.45219 

.82546 

.46688 

.87574 

.48172 

.92947 

.49673   .987XX) 

47 

48 

.45244 

.82627 

.46712 

.87661 

!  .48197 

.93040  ;  .49698 

.98799 

48 

49 

.45268 

.82709 

.46737 

.87748 

.48222 

.93133   .497'23 

.98899 

49 

50 

.45292 

.82790 

.46762 

.87834 

.48247 

.93226 

.49748 

.98998 

50 

51 

.45317 

.82871 

.46786 

.87921 

.48272 

.93319 

.49773 

.99098 

51 

52 

.45341 

.82953 

.46811 

.88008 

.48297 

.93412   .49799 

.99198 

52 

53 

,45365 

.83034 

.46836 

.88095 

.46322 

.93505   .49824 

.99298 

53 

54 

.45390 

.&3116 

.46860 

.88183 

!  .48347 

.93598  |i  .49849 

.99398 

54 

55 

.45444 

.83198 

!  .46885 

.88270 

.48372 

.93692  !  .49874 

.99498 

55 

56 

.45439 

.83280 

i  .46909 

.88357 

.48396 

.93785  :  .49899   .99598 

56 

57 

.45463 

.83362 

.46934 

.88445 

j  .48421   .93879  i  .49924 

.99698 

57 

58 

.45487 

.83444 

.46959 

.88532 

.48446 

.93973 

.49950 

.99799 

58 

59 

.45512 

.83526 

.46983 

.88620 

.48471 

.94066  I  .49975 

.99899 

59 

60   .45536   .83608 

1  .47008 

.88708 

.48496   .94160  1  .50000  1.00000 

60 

484 


TABLE  XXIX.- NATURAL  VERSED  SINES  AND  EXTERNAL    SECANTST 


6 

-   ) 

• 

i° 

6 

2° 

6 

3° 

Vers. 

Ex  .se 

Vers.  ; 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

i 

0 

.50000 

.00000 

.51519  i 

1.06267 

.53053 

1.13005  | 

.54601 

1.20269 

0 

1 

50025 

00101 

.51544 

1.06375 

.53079 

1.13122 

.54627 

1.20395 

1 

2 

.50050 

.00202 

.51570 

1.06483 

.53104 

1.13239 

.54653 

.20521 

2 

3 

50076 

.001303 

.51595 

1.06592 

.53130 

1.13,356 

.54679 

.20647 

3 

4 

.50101 

00404 

.51621 

1.06701 

.53156 

1.13473 

.54705 

.20773 

4 

5 

.50126 

.00505 

.51646 

1.06809 

.53181 

1.13590 

.54731 

.20900 

5 

(1 

50151 

1.00607 

.51672 

1.06918 

.53207 

1.13707 

.54757 

.21026 

6 

7 

.50176 

1.00708 

.51697 

1.07027 

.53233 

1.13825 

.54782 

.21153 

r* 

,s 

50202 

1.00810 

.51723 

1.07137 

.53258 

1.13942 

.54808 

.21280 

8 

9 

.50227 

1.00912 

.51748 

1.07246 

.53284 

1.14060 

.54834 

.21407 

9 

10 

.50252 

1.01014 

.51774 

1.07356 

.53310 

1.14178 

.54860 

.21535 

10 

11 

.50277 

1.01116 

.51799 

1.07465 

.53330 

1.14296 

.54886 

.21662 

11 

12 

.50303 

1.01218 

.51825 

1.07575  i 

.53361 

1.14414 

.54912 

.21790 

12 

13 

.50328 

1.01320 

.51850 

1.07685  i 

.53387 

1.14533 

.54938 

.21918 

13 

14 

.50353 

1.01422 

.51876 

1.07795 

.53413 

1.14651 

1  .54964 

.22045 

14 

15 

.50378 

1.01525 

.51901 

1.07905 

.53439 

1.14770 

.54990 

.22174 

15 

Ki 

.50404 

1.01628 

.51927 

1.08015 

.53464 

1.14889 

.55016 

.22302 

16 

17 

.50429 

1.01730 

.51952 

1.08126 

.53490 

1.15008 

.55042 

.22430 

17 

18 

.50454 

1.01833 

.51978 

1.08236 

.53516 

1.15127 

.55068 

.22559 

18 

19 

.50479 

1.01936 

.52003 

1.08347 

.53542 

1.15246 

.55094 

.22688 

19 

80 

.50505 

1.02039 

.52029 

1.08458 

.53567 

1.15366 

.55120 

.22817 

20 

'.31 

50530 

1.02143 

.520,54 

1.08569 

.53593 

1.15485 

.55146 

.22946 

21 

2-2 

.50555 

1.02246 

.52080 

1.08680 

.53619 

1.15605 

.55172 

.23075 

22 

93 

.50581 

1.02349 

.52105 

1.08791 

.53645 

1.15725 

.55198 

.23205 

23 

21 

.50606 

1.02453 

.52131 

1.08903 

.53670 

1.15845 

.55224 

.23334 

24 

25 

.50631 

1.02557 

.52156 

1.09014 

.53096 

1.15965 

.55250 

.23464 

25 

26 

50656 

1.02661 

.52182 

1.09126 

.53722 

1.16085 

.55276 

.23594 

26 

27 

.50682 

1.02765 

.52207 

1.09238 

.53748 

1.16206 

.55302 

.23724 

27 

28 

.50707 

1.02869 

.52233 

1.09350 

.53774 

1.16326 

.55328 

.23855 

28 

•.".» 

50732 

1.02973 

.52259 

1.09462 

.53799 

1.16447 

.55354 

.23985 

29 

30 

.50758 

1.03077 

.52284 

1.09574 

.53825 

1.16568 

.55380 

.24116 

30 

•\\ 

50783 

1.03182 

.52310 

1.09686 

.53851 

1.16689 

.55406 

.24247 

31 

32 

.50808 

1.03286 

.52335 

1.09799 

.53877 

1.16810 

.55432 

.24378 

32 

33 

.50834 

1.03391 

.52361 

1.09911  ! 

.53903 

1.16932 

.55458 

.24509 

33 

34 

.50859 

1.03496 

.52386 

1.10024 

.53928 

1.17053 

.55484 

.24640 

34 

35 

.50884 

1.03601 

.52412 

1.10137 

..53954 

1.17175 

.55510 

.24772 

35 

36 

50910 

1.03706 

.52438 

1.10250 

.53980 

1.17297 

.55536 

.24903 

36 

37 

.50935 

1.03811 

.52463 

1.10363 

.54006 

1.17419 

.55563 

.25035 

37 

88 

50960 

1.0391(5 

.52489 

1.10477 

.54032 

1.17541 

.:-!5589 

25167 

38 

39 

.50986 

1.04022 

.52514 

1.10590 

.54058 

1.17663 

.55615 

.25300 

39 

40 

.51011 

1.04128 

.52540 

1.10704 

.54083 

1.17786 

.55641 

.25432 

40 

41 

.51036 

1.04233 

.52566 

1.10817 

.54109 

1.17909 

.55667 

.25565 

41 

42 

.51062 

1.04339 

.52591 

1.10931 

.54135 

1.18031 

.55693 

.25697 

42 

W 

.51087 

1.04445 

.52617 

1.11045 

.54161 

1.18154 

.55719 

.25830 

43 

It 

.51113 

1.04551 

.52642 

1.11159 

.54187 

1.18277 

.5^745 

.25963 

44 

46 

.51138 

1.04658 

.52668 

1.11274 

.54213 

1.18401 

.55771 

1.26097 

4,-> 

46 

.51163 

1.04764 

.52694 

1.11388 

.54238 

1.18524 

.55797 

1.26230 

46 

47 

.51189 

1.04870 

.52719 

1.11503 

.54264 

1.18648 

.55823 

1.26364 

47 

is 

.51214 

1.04977 

|  .52745 

1.11617 

.54290 

1.18772 

.55849 

1.26498 

48 

IS 

.51239 

1.05084 

.52771 

1.11732 

.54816 

1.18895 

.55876 

1.26632 

49 

50 

.51265 

1.05191 

.52796 

1.11847 

.54342 

1.19019 

.55902 

1.26766 

50 

51 

.54290 

1.05298 

.52822 

1.11963 

.54368 

1.19144 

.55928 

1.26900 

51 

52 

.51316 

1.05405 

.52848 

1.12078 

i  .54394 

1.19268 

.55954 

1.27035 

]52 

53 

.51341 

1.05512 

.52873 

1.12193 

!  .54420 

1.19393 

.55980 

1.27169 

J53 

54 

.51366 

1.05619 

.52899 

1.12309 

.;V1H<) 

1.19517 

.56006 

1.27304 

54 

55 

.51392 

1.05727 

.52924 

1.12425 

.5-1471 

1.19642 

.56032 

1.27439 

155 

66 

.51417 

1.05835 

.52950 

1.12540 

.54497 

1.1976? 

.56058 

1.27574 

56 

57 

.51443 

1.05942 

.52976 

1.12657 

.54523 

1.19892 

.56084 

1.27710 

57 

68 

i  .51468 

1.06050 

.53001 

1.12773 

j  .54549 

1.20018 

.56111. 

1.27845 

153 

,VJ 

.51494 

1.06158 

.53027 

1.12889 

.54575 

1.20143 

.56137 

1.27981 

59 

GO 

1  .51519 

1  1.06267 

.53053 

1.13005 

.54601 

1.20269 

.56163 

1.28117 

60 

485 


TABLE  XXIX. -NATURAL  VERSED  SINES  AND   EXTERNAL  SECANTS. 


' 

64°          65°          66C          67° 

' 

Vers. 

Ex.  sec.  I 

Vers.  Ex.  sec. 

Vers.  :  Ex.  sec.  Vers.  Ex.  sec. 

0 

.56163  i  1.28117  i 

.57738   1.36620  '  .59326  |  .45859   .60927   1.559:30 

(i 

1 

.56189  i  1.28253   .57765   1.36768   .59*53  !  .46020   .60954   1.56106   1 

') 

.56215  i  1.28390  i  .57791   1.36916  !  .59379   .46181  i  .60980  1.56282 

2 

3 

.56241 

1.28326  M  .57817   1.37064  1  .59406   .46342  il  .61007  !  1.56458 

3 

4 

.56267 

1.28663  |  .57844  1.37212  i  .594:33   .46504 

.61034   1.56634 

4 

5 

.56294 

1.28800   .57870  1.37361 

.59459  ;  .46665 

.61061   1.56811 

5 

8 

.56320 

1.28937  !  .57896   1.37509 

.59486  i  .46827 

.61088  i  1.56988 

6 

7 

.56346 

1.39074  i  .57923   1.37658  i 

.59512  i  .46989 

.61114  i  1.57165 

7 

8 

.56372 

1.29211  j 

.57949   1.37808 

.59539 

.47152 

.61141   1.57:342 

8 

9 

.56398 

1.29349 

.57976 

1.37957 

.59566 

.47314 

.61168   1-57520 

9 

10 

.56425 

1.29487 

.58002 

1.38107 

.59592 

.47477  'i 

.61195  1.57698 

10 

11 

.56451 

1.29625 

.58028 

1.38256   .59619 

.47640 

.61222  1.57876 

11 

12 

.56477 

1-29763 

.58055 

1.38406  i  .59645 

.47'804 

.61248   1>)M)54 

12 

18 

.56503 

1.29901 

.58081 

1.38556  i 

.59672 

.47967   .61275   1.58233 

18 

14 

.56529 

1.30040 

.58108  1.38707  ; 

.59699 

.48131   .61302   1.58412 

U 

15 

.56555 

1.30179 

.581134 

1.38857 

.59725 

.48295  i  .61329   1.58591 

16 

16 

.56582 

1.30318 

.58160 

1.39008  !  .59752   .48459  i  .61356   1.58771 

1C, 

17 

.56608 

1.30457 

.58187 

1.39159  i  .59779 

.48624   .61383   1.58950 

17 

18  .56(534 

1.30596  j 

.58213 

1.39311  ; 

.59805 

.48789  |  .61409  1.59130 

18 

19!  .566(50 

1.30735  ; 

.58240  i  1.39462 

.59832 

.48954   .61436   1.59311 

1!) 

20 

.56687  1  1.30875 

.58266 

1.39614  j 

.59859   .49119   .61463   1.59491 

20 

21 

.56713  1.31015 

.58293 

1.39766  ' 

,59885 

.49284 

.61490  1.59672 

21 

22 

.56739 

1.31155 

.58319 

1.3.1918  ; 

.59912   .49450   .61517 

1.59853 

22 

28 

.56765 

1.31295 

.58345 

1.40070   .599:38 

.49616  ;  .61544 

1.60035 

28 

24 

.56791 

1.31436 

.58372 

1.40222 

.59965 

.49782  :  .61570 

1.60217 

24 

»! 

.56818  !  1.31576 

.58398 

1.40375 

.59992   .49948 

.61597 

1.60399 

25 

-,'i; 

.56844 

1.31717 

.58425 

1.40528 

.60018 

.50115 

.61624 

1.60581 

26 

27 

.56870 

1.31858 

.58451 

1.40681 

.60045 

.50282 

.61651 

1.60763 

27 

88 

.56896 

1.31999 

.58478 

1.40835 

.60072   .5044!* 

.61678 

1.60946  2S 

20 

.56923 

1.32140 

.58504 

1.40988 

.60098 

.50617 

.61705 

1.61129  29 

30 

.56949 

1.32282 

.58531 

1.41142 

.60125 

.50784 

.61732 

1.61313  30 

81 

.56975 

1.32424 

.  58557 

1.41296 

.60152 

.50952 

.61759 

1.61496 

31 

33 

.57001 

1.32566 

.58584 

1.41450 

.60178 

.51120 

.61785 

1.61680 

32 

33 

.57028 

1.327'08 

.58610 

1.41605 

.60205 

.51289 

.61812 

1.61864 

88 

34  1  .57054 

1.32850 

.58637 

1.41760 

.60232 

.51457 

.61839 

1.62049 

m 

35 

.57080 

1.32993 

.58663 

1.41914 

.60259 

.51626 

.61866 

1.62234 

35 

36 

.57106 

1.33135  i 

.58690 

1.42070 

.60285   .51795 

!  61893 

1.62419 

36 

37 

.57133 

1.33278 

.58716 

1.42225 

.60312 

.51965 

.61920 

1.62604 

37 

38 

.57159 

1.33422 

.58743 

1.42380 

.60339 

.521:34 

.61947 

1.62790 

38 

39 

.57185 

1.33565 

.58769 

1.42536  , 

.60365 

.52304  : 

.61974 

1.62976 

39 

lit 

.57212 

1.33708 

.58796 

1.42692 

.60392 

.52474 

.62001 

1.63162 

40 

11 

.57238 

1.33852  : 

.58822 

1.42848 

.60419 

.52645 

.62027 

1.63348 

41 

4S 

.57264 

1.33996  i 

.58849 

1.43005 

.60445 

.52815 

.62054 

.63535 

42 

43 

.57291 

1.34140  i 

.58875 

1.43162 

.60472 

.52986  , 

.62081 

.63722 

•U 

44  .57317 

1.34284 

.58902 

1.43318 

.60499 

.53157 

.62108 

.63909 

44 

45 

.57343 

1.34429  ! 

.58928 

1.43476 

.60526 

.5,3329 

.62135 

.64097 

45 

46 

.57369 

1.3-1573 

.58955 

1.43633 

.60552 

.53300 

.62162 

.64285 

40 

47  .57396 

1.34718  i 

.58981  I  1.43790 

.60579 

.53672 

.62189 

.64473 

47 

48  i  .57422  !  1.34863  i 

.59008   1.43948 

.60606 

.53845 

.62216 

.64662 

48 

49 

.57448 

1.35009 

.59034  i  1.44106 

.60633 

.54017  i 

.62243 

.64851 

49 

50 

.57475 

1.35154 

.59061 

1.44264 

.60659 

.54190  . 

.62270 

.65040 

50 

51 

.57501 

1.35300  1 

.59087 

1.44423 

.60686 

.54363 

.62297 

.65229 

51 

58 

.57527  i  1.35446  , 

.59114   1.44582 

.60713 

.54536 

.62324 

.65419 

52 

58 

.57554  i  1.35592  , 

.59140  i  1.44741 

.60740 

.54709 

.62351 

.65609  153 

r,i 

.57580  1.35738 

.59167   1.44900 

.60766 

.54883 

.62378 

.65799  54 

55 

.57606   1.3.-SS5 

.59194   1.45059 

.60793 

.55057 

.62405 

.65989  .53 

56  .57'633 

1.36031 

.59220 

1.45219 

.60820 

.55231 

.62431 

.66180  56 

57 

.57659 

1.36178 

.59247 

1.45378 

.60847 

.55405 

.62458 

.66371  57 

ns 

.57685  I  1.36325 

.59273 

1.45689 

.60873 

.55580 

.62485 

.66563  58 

59 

.57712  1.36473 

.59300 

1.45699 

.60900 

.  55755 

.62512 

1.66755  !5'J 

(SO 

.57738  !  1.36620  i 

.59326 

1.45859 

.60927 

.55930 

.62539 

1.66947  100 

TABLE    XXIX.— NATURAL  VERSED  SINES  AND    EXTERNAL  SECANTS. 


/ 

68° 

69°          70° 

71. 

;  / 

Vers. 

Ex.  sec. 

Vers.  Ex.  sec. 

,  Vers. 

I 

Ex.  sec. 

Vers.  Ex.  sec. 

0 

.62539 

1.66947 

.64163 

1.7'9043 

.65798 

1.92380 

.67443  !  2.07155 

( 

1 

.62566 

1.67139 

.64190 

1.79254 

.65825 

1.92614 

.67471  !  2.07415 

• 

2 

.62593 

1.67333 

.64218 

1.79466 

i  .65853 

1.92849 

.67498 

2.07-67-5 

< 

3 

.62620 

1.67525 

.64245 

1.79679 

.65880 

1.93083 

.67526 

2.07936 

j 

4 

.62647 

1.67718 

.64272 

1.79891 

!  .65907 

1.93318 

.67553 

2.08197 

i 

5 

.6267'4 

1.67911 

.64299 

1.80104 

.65935 

1.93554 

.67581 

2.08459 

{ 

6 

.62701 

1.68105 

.64326 

1.80318 

.65962 

1.93790 

.67608 

2.08721 

1 

7 

.62723 

1.68299 

.64353 

1.80531 

.65989 

1.94026 

.67636 

2.08983 

\ 

8 

.62755 

1.68494 

.64381   1.807'46 

.66017 

1.94263 

.67663 

2.09246 

j 

9 

.62782  1.68689 

.64408 

1.80960   .66044 

1.94500 

.67691 

2.09510 

c 

10 

.62309 

1.68=84 

.64435 

1.81175 

.66071 

1.94737 

.67718 

2.09774 

10 

11 

.62836 

1.69079 

.64462 

1.81390 

.66099 

1.94975 

.67746 

2.10038 

11 

12 

.62863 

1.69275 

.64489 

1.81605 

.66126 

1.95213 

.67773 

2.10303 

12 

13 

.62890 

1.69471 

.64517 

1.81821 

i  .66154 

1.95452 

.67801 

2.10568 

13 

14 

.62917 

1.69667 

.64544 

1.82037 

i  .66181 

1.95691 

.67829 

2.10834 

14 

15 

.62944 

1.69864 

.64571 

1.82254 

.66208 

1.95931 

.67856 

2.11101 

15 

16 

.62971 

1.70061 

.64598 

1.82471 

.66236 

1.96171 

.67884 

2.11367 

16 

17 

.62998 

1.70258 

.64625 

1.82688 

.66263   1.96411 

.67911 

2.11635 

17 

18 

.63025 

1.70455 

.64653 

1.82906 

.66290  i  1.96652 

.67939 

2.11903 

18 

19 

.63052 

1.70653 

.64680 

1.83124 

.66318 

1.96893 

.67966 

2.12171 

19 

20 

.63079 

1.70851 

.64707 

1.8a342 

.66345 

1.97135 

.67994 

2.12440 

20 

21 

.6310S 

1.71050 

.64734 

1.83561 

.66373 

1.97377 

.68021 

2.12709 

21 

22 

.63133 

1.71249 

.64761 

1.83780 

.66400 

1.97619 

.68049 

2.12979  i22 

23 

.63161 

1.71448 

.(54789 

1.83999 

!  .66427 

1.97862 

.68077 

2.13249 

23 

24 

.63188 

1.71647 

.64816 

1.84219  II  .66455 

1.98106 

.68104 

2.13520 

24 

25 

.63215 

1.71847 

.64843 

1.84439 

!  .66482 

1.98349 

.68132 

2.13791 

25 

26 

.63242 

1.72047  i 

.64870 

1.84659 

.66510 

1.98594 

.68159 

2.14063 

26 

27 

.63269 

1.72247  ; 

.64898  1.84880 

.66537 

1.98838 

.68187 

2.14335 

27 

28 

.63296 

1.72448  i 

.64925   1.85102 

.66564 

1.99088 

.68214 

2.14608 

28 

29 

.63323 

1.72649  ! 

.64952  I  1.85323 

.66592 

1.99329 

.68242 

2.14881 

2( 

30 

.63350 

1.72850 

.64979 

1.85545 

.66619 

1.99574  | 

.68270 

2.15155 

30 

31 

.63377 

1.73052 

.65007 

1.85767 

.66647 

1.99821 

.68297 

2.15429 

31 

32 

.63404 

1.73254 

.65034 

1.85990 

.66674 

2.00067 

.68325 

2.15704 

32 

33 

.63431 

1.73456 

.65061 

1.86213 

.66702 

2.00315 

.68352 

2.15979 

33 

34 

.63458 

1.73659 

.65088 

1.86437 

i  .66729 

2.00562 

.68380 

2.16255 

34 

35 

.63485 

1.73862 

.65116 

1.86661 

1  .66756 

2.00810 

.68408 

2.16531 

35 

36 

.63512 

1.74065 

.65143 

1.86885 

.66784 

2.01059 

.68435 

2.16808 

36 

37 

.63539   1.74269  i 

.65170 

1.87109 

.66811  2.01308 

.68463 

2.17085 

37 

38 

.63566   1.74473  j 

.65197 

1.87334 

.66839  2.01557 

.68490 

2.17363 

38 

39 

.63594  1  1.74677 

.65225 

1.87560 

.66866  2.01807 

.68518 

2.17641 

39 

40 

.63621 

1.74881 

.65252 

1.87785 

.66894  j  2.02057 

.68546 

2.17920 

40 

41 

.63648 

1.7.5036 

.65279 

1.88011 

.66921  2.02308 

.68573 

2.18199 

41 

42 

.68675 

1.75292  j 

.65306 

1.88238 

.66949  i  2.02559 

.6QP01 

2.18479 

42 

43 

.63702 

1.75497  i 

.65334 

1.88465 

.6697'6 

2.02810 

.68628 

2.18759 

43 

44 

.63729 

1.75703  ; 

.65361 

1.88692 

.67-003 

2.03062 

.68656 

2.19040 

44 

S 

.63756 

1.7'5909 

.65388 

1.88920 

.67031   2.03315 

.68684 

2.19322 

45 

[6 

.63783 

1.76116 

.65416 

1.89148 

.67-058  2.03568 

.68711 

2.19604 

46 

[7 

.63810 

1.76323 

.65443 

1.89376  ii  .67086  2.03821 

.68739 

2.19886 

47 

48 

.63838 

1.76530  1 

.6547'0   1.89605   .67113  i  2.04075 

.68767 

2.20169 

48 

49 

.63865 

1.767'37 

.65497 

1.89834  i;  .67141 

2.04329 

.68794 

2.20453 

49 

50 

.63892 

1.76945 

.65525 

1.90063  !|  .67168 

2.04584 

.68822 

2.20737 

50 

51 

.63919 

1.77154  ; 

.65552 

1.90293  i  .67196 

2.04839 

.68849 

2.21021 

51 

52 

.63946 

1.77362  i 

.65579 

1.90524   .67223 

2.05094 

.68877 

2.21306 

52 

53 

.63973 

1.7757* 

.65607 

1.90754   .67'251 

2.05350 

.68905 

2.21592 

53 

54 

.64000 

1.77780  i 

.65634 

1.90986  |  .67278 

2.05607 

.68932 

2.21878 

54 

>5 

.64027 

1.77990  i 

.65661 

1.91217   .67306 

2.05864 

.68960 

2.22165 

55 

56 

.64055 

1.78200  : 

.65689 

1.91449   .67333 

2.06121 

.68988 

2.22452 

56 

57 

.64082 

1.78410  ; 

.65716 

1.91681 

.67361   2.06379 

.69015 

2.22740 

57 

58 

.64109 

1.78621  ! 

.65743 

1.91914 

.67388  2.06637 

.69043 

2.23028 

58 

59 

.64136 

1.78832  ! 

.65771 

1.92147 

.67416  2.06896 

.69071 

2.23317 

59 

60 

.64163 

1.79043 

.65798 

1.92380 

.67443  i  2.07155  1 

.69098 

2.23607 

30 

487 


TABLE  XXIX.— NATURAL  VERSED  SINES  AND    EXTERNAL  SECANTS. 


< 

72°          73°          74°          75° 

' 

Vers. 

Ex.  sec.  \  Vers. 

Ex.  sec.: 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.69098 

2.23607  'i  .70763  i  2.42030 

.72436 

2.62796  :  .74118 

2.86370 

<» 

1 

.69126 

2.23897  1!  .70791  i  2.42356 

.72464 

2.63164  ;  .74146 

2.86790 

5 

g 

.69154 

2.24187  !  .70818  2.42683 

.72492  1  2  63533  1  .74174 

2.87211 

2 

3 

.69181 

2.24478 

.70846 

2.43010 

.72520 

2.03903  1  .74202 

2.87633 

3 

4 

.69209 

2.24770 

.70874 

2.43337 

.72548 

2.64274  ;  .74231 

2.88056 

4 

5 

.69237 

2.25062 

.70902  ;  2.43666 

.72576  j  2.64045 

.74259 

2.88479 

5 

0 

,69264 

2.25355 

.70930 

2.43995 

.72604 

2.65018 

.74287 

2.88S01 

6 

7 

.69292 

2.25648 

.70958 

2.44324 

.72632 

2.65391 

.74315 

2.89330 

7 

8 

.69320 

2.25942 

.70985 

2.44655 

.72660  1  2.65705 

.74343 

2.89750 

8 

9 

.69347 

2  26237 

.71013 

2.44986 

.72688 

2.66140  i 

.74371 

2.90184 

9 

10 

.69375 

2.26531 

.71041 

2.45317 

.72710 

2.66515  ; 

.74399 

2.9CG13 

10 

11 

.69403 

2.26827 

.71069 

2.45650 

.72744 

2.60892 

.74427 

2.91042 

11 

19 

.69430 

2.27123 

.71097 

2.45983 

.72772 

2.67269  ^ 

.74455 

2.91473  j  12 

13 

.69458 

2.27420 

.71125 

2.46316 

.72800 

2.67647 

.74484 

2.91904  !13 

11 

.69486 

2.27717 

.71153 

2.46651 

.72828 

2.  68025  J  .74512 

2.92337  ;14 

15 

.69514 

2.28015 

.71180 

2.46986 

.72850 

2.68405   .74540 

2.92770  i  15 

10 

.69541 

2.28313 

.71208  2.47321 

.72884 

2.68785  : 

.74568 

2.93204  16 

17 

.69569 

2.28612 

.71236  2.47658 

.72912 

2.69167 

.74596 

2.93640  17 

IS 

.69597 

2.28912 

.71204 

2.47995 

.72940 

2.69549 

.74624 

2.94076  [18 

1!) 

.69624 

2.29212 

.71292 

2.48333 

.72968 

2.69931 

.74652 

2.94514  119 

20 

.69652 

2.29512 

.71320 

2.48671 

.72996 

2.70315  i 

.74680 

2.94952 

20 

21 

.69680 

2.29814 

.71348 

2.49010 

.73024 

2.70700 

.74709 

2.95892 

•21 

28 

.69708 

2.30115 

.71375 

2.49350 

.73052 

2.71085 

.74737 

','  .  1)5832  '  22 

23 

.69735 

2.30418 

.71403 

2.49691 

.73080 

2.71471 

.747'65 

2!  1)6274  23 

24 

.69763 

2.30721 

.71431 

2.50032 

.73108 

2.71858 

.747'93 

2.90716  124 

25 

.69791 

2.31024 

.71459 

2.50374 

.73136 

2.72246 

.74821 

2.97160 

35 

•JO 

.69818 

2.31328 

.71487 

2.50716 

.73164 

2.72635 

.74849 

2.97604 

2C 

27 

.69846 

2.31633 

.71515 

2.51060 

.73192 

2.73024 

.74878 

2.98050 

27 

;j,S 

.69874 

2.31939 

.71543 

2.51404 

.73220 

2.73414  ' 

.74906 

2.98497 

28 

i;!) 

.69902 

2,32244 

.71571 

2.51748 

.73248 

2.73806 

.74934 

•-'.US'.  Ml 

29 

80 

.69929 

2.32551 

.71598 

2.52094 

.73276 

2.74198 

.74962 

2.99393 

30 

31 

.69957 

2.32858 

.71626 

2.52440 

.73304 

2.74591 

.74990 

2.99843 

31 

3-J 

.69985 

2.33166 

.71654 

2.52787 

.73332 

2.74984 

.75018 

3.0021)3 

82 

00 

.70013 

2.33474 

.71682 

2.53134 

.73360 

2.75379 

.75047 

3.00745 

38 

34 

.70040 

2.33783 

.71710 

2.53482 

.73388 

2.75775 

.75075 

3.01198 

84 

86 

.70068 

2.34092 

.71738 

2.53831 

.73416 

2.76171 

.75103 

3.01052 

85 

30 

.70096 

2.34403 

.71766 

2.54181 

.73444  2.70508 

.75131 

3.02107 

86 

37 

.70124 

2.34713 

.71794 

2.54531 

.73472  2.76960 

.7'5159 

3.02563 

87 

38 

.70151 

2.35025 

.71822  !  2.54883 

.73500 

2.77365 

.75187 

3.03020 

88 

89 

.70179 

2.35336 

.71850  2.55235 

.73521) 

2.77765 

.75216 

3.03479 

89 

40 

.70207 

2.35649 

.71877 

2.55587 

.73557 

2.78166 

.75244 

3.03938 

fi) 

41 

.70235 

2.35962 

.71905 

2.55940 

.73585 

2.78568 

.75272 

3.04398 

41 

49 

.70263 

2.36276 

.71933 

2.56294 

.73613 

2.7897'0 

.75300 

3.04800 

42 

4:5 

.70290 

2.36590 

.71961 

2.56649 

.73641 

2.79374 

.75328 

8.05322 

13 

It 

.70318 

2.36905 

.71989 

2.57005 

.73669 

2.79778 

.75356 

3.05786  J44 

45 

.70346 

2.37221 

.72017 

2.573G1 

.73697 

2.80183 

.75385 

3.06251 

45 

46 

.70374 

2.37537 

.72045 

2.57718 

.73725 

2.80589 

.75413 

3.06717 

40 

47 

.70401 

2.37854 

.72073 

2.58076 

.73753 

2.80996 

.75441 

3.07184 

47 

48 

.70429 

2.38171 

.72101 

2.58434 

.73781 

2.81404 

.75469 

3.07652 

48 

49 

.70457 

2.38489 

.72129 

2.58794 

.73809 

2.81813 

.75497 

3.08121 

49 

50 

.70485 

2.38808 

.72157 

2.59154 

.73837 

2.82223 

.75526 

3.08591 

50 

51 

.70513 

2.39128 

.72185 

2.59514 

.73865 

2.826a3 

.75554 

3.09063 

51 

.70540 

2.39448 

.72213 

2.59876 

.73893  2.83045 

.75582 

3.09535 

52 

53 

.70568 

2.39768 

.72241 

2.60238 

.73921  2.&S457 

J5G10 

3.10009 

§8 

54 

.70596 

2.40089 

.72269  !  2.60601 

.73950 

2.83871 

^5639 

3.10484 

54 

56 

.70624 

2.40411 

.72296 

2.60965 

.73978 

2.84285 

.75667 

3.10960 

55 

50 

.70652 

2.40734 

.72324 

2.61330 

.74006 

2.84700  i 

.75695 

3.11437 

56 

57 

.70679 

2.41057 

.72352 

2.61695 

.74034 

2.85116 

.75723 

3.11915 

57 

58 

.70707 

2.41381 

.72380 

2.62061 

.74062 

2.85533  1 

.75751 

3.12394 

58 

59 

.70735 

2.41705 

.72408 

2.62428 

.74090 

2.85951  ! 

.75780 

3.12875 

59 

60 

.70763 

2.42030 

.72436  2.6279G 

.74118 

2.86370  i 

.75808 

3.13357 

60 

488 


TABLK  XXIX.— NATURAL  VERSED  SINES  AND  EXTERNAL    SECANTS 


76°                       77°                       78° 

79° 

Vers.     Ex.  sec.  i  Vers.     Ex.  sec.  ;    Vers.    j  Ex.  sec. 

Vers. 

Ex.  sec. 

0     .75808     3.1335?    i   .77505     3.44541  |j   .79209 

3.  80973 

i   .80919 

4.24084 

0 

li   .75836  :  3.13839       .77533 

3.45102 

.79237 

3.816&3 

i   .80948 

4.24870 

1 

2  ]   .75864  I  3.14323       .77562 

3.45664 

.79266 

3.82294 

!   .80976 

4.25658 

2 

3 

.75892 

3.14809  i     .77590     3.46228 

.79294 

3.82956 

i   .81005 

4.26448 

8 

4 

.75921 

3.15295  i     .77618 

3.46793 

.79323 

3.83621 

i   .81033 

4.27241 

-! 

5     .75W.) 

3.15782  i     .77647 

3.47360 

.79351 

3.84288 

;   .81062 

4.28036 

5 

6  1   .75977 

3.16271 

.7767o 

3.47928 

.79380 

3.84956 

.81090 

4.28833 

6 

.76005 

3.16761 

.77703 

3.48498 

.79408 

3.85627 

i   .81119 

4.29634 

8     .76031 

347252 

.77732 

3.49069 

.79437 

3.86299 

;    .81148 

4.  3043(5 

8 

y,  .76062 

3.17744 

.77760 

3.49642 

.79465 

3.86973  !     .81176 

4.31241 

9 

10  1   .76090 

3.18238 

.77788 

3.50210 

.79493 

3.87649 

'   .81205 

4.32049 

Hi 

11     .76118 

3.1S733    j   .77817 

3.50791 

.79522 

3.88327 

!   .81233 

4.32859 

1! 

12     .76147 

3.19228 

:   .77845 

3.51368 

.79550 

3.89007 

i   .81262 

4.33671  !12 

13     .76175 

3.19725 

.77S74     3.51947 

.79579 

3.89689 

"  .81290 

4.34486 

13 

14     .76-303 

3.20224 

.77902 

3.52527 

.79607 

3.90373 

i   .81319     4.35304 

11 

15     .76231 

3.20723 

.77930 

3.53109 

.79636 

3.91058 

.81348 

4.36124    15 

16     .76260 

3.21224 

.77959 

3.53692 

.79664 

3.91746 

'•   .81376 

4.36947  :16 

17  i   .7<ttKS 

3.21726 

.77987 

3.54277  i     .79693 

3.92436 

.81405 

4.37772  117 

18     .76316 

3.22229 

.78015 

3.54863  !     .79721 

3.93128 

.81433     4.38600  il8 

19|*.  76344 

3.22734 

.78044 

3.5,5451 

.79750 

3.93821 

j   .81462     4.39430    19 

20 

.70878 

3.23239 

.7'807'2 

3.56041 

.79778 

3.94517 

1   .81491     4.40263    20 

-.'] 

.76401 

3.23746 

.78101 

3.56632 

.79807 

3.95215 

.81519 

4.41099    21 

22 

.76429 

3.24255 

.78129 

3.57224 

.79835 

3.95914 

.81548 

4.41937    22 

•>.; 

.76158 

8,24764 

.78157 

3.57819 

.79864  :  3.96616 

.81576 

4.42778 

;.':; 

24 

.76486 

3.25275 

.78186 

3.58414 

.79892      3.97320 

.81605 

4.43622 

24 

;.':> 

.76514 

3.35787 

.78214 

3.59012 

.79921      3.98025 

.81633 

4.44468 

25 

•jr. 

.76543 

3.26300 

.78242 

3.59611 

..79949 

3.98733 

;   .81662 

4.45317 

26 

.••• 

.76571 

3.26814 

.78271 

3.60211 

.7997'8 

3.99443 

.81691 

4.46169 

87 

ys 

.76599 

3.27330 

.78299 

3.60813 

.80006 

4.00155 

.81719 

4.47023 

28 

29 

.76627 

3.27847 

.78328 

3.61417 

.80035 

4.00869 

1   .81748 

4.47881 

-,".) 

80 

.76655 

3.28366 

.78356 

3.62023 

.80063 

4.01585 

.81776 

4.48740 

80 

31 

.76684 

3.28885 

.78384 

3.62630 

.80092 

4.02303 

.81805 

4.49603 

81 

82 

.76712     3.29406 

.78413 

3.63238 

.80120 

4.03024 

.81834 

4.50468 

32 

•',:', 

.76740 

3.29929 

.78441 

3.63849 

.80149 

4.03746 

.81862 

4.51337 

38 

34 

.76769 

3.30452 

.78470 

3.64461 

.80177     4.04471 

.81891 

4.52208 

34 

35 

.76797 

3.30977 

.78498 

3.65074 

.80206 

4.05197 

.81919 

4.53081 

85 

36 

.76825 

3.31503 

.78526 

3.65690 

.80234 

4.05926 

.81948 

4.53958  |36 

:>; 

.70854 

3.32031        .78555 

3.66307 

.80263 

4.06657 

.81977 

4.54837  !37 

38 

.76882 

3.32560 

.78583 

3.66925 

.80291 

4.07'390 

.82005 

4.55720    38 

39 

.76910 

3.33090 

.78612 

3.67545 

.80320 

4.08125 

.82034 

4.56605    39 

in 

.76938 

3.33622       .78640 

3.6S167 

.80348 

4.08863 

.82063 

4.57493    40 

!! 

.76967 

3.34154 

.78669 

3.68791 

.80377 

4.09602 

.82091 

4.58383    41 

42     .76995 

3.34689 

.78697 

3.69417 

.80405 

4.10344 

.82120 

4.59277    42 

43     .77'0r>3 

3.35224 

.78725 

3.70044 

.804:34 

4.11088 

.82148 

4.6017'4    43 

44     .77052 

3.35761 

.78754 

8.7Q678 

.80462 

4.11835 

.82177 

4.61073 

II 

45 

.77080 

3.36299 

.78782 

3.71303 

.80491 

4.12583 

.82206 

4.61976 

15 

46 

.77108 

3.36839 

[78811 

3.71935 

.80520 

4.13334 

.82234 

4.62881 

]<; 

47    .77137 

3.37380 

.78839 

3.72569 

.80548 

4.14087 

.82263 

4.63790    47 

48  i   .77165 

3.37923 

.78868 

3.73205 

.80577 

4.14842 

.82292      4.64701   ;48 

49 

.77193 

3.38466 

.78896 

3'  78843 

.80605 

4.15590 

.82320      4.65616  i49 

50 

.77222 

3.39012 

.78924 

3.7'4482  j 

.80634     4.16359 

.82349 

4.66533    50 

M 

.77250 

3.39558 

.78953 

3.75123 

.80662     4.17121 

.82377 

4.67454  1  51 

52 

.77'27'8 

3.40106 

.78981 

3.75766 

.80691     4.17886 

.82406 

4.68377  152 

53 

.77307 

3.406*6 

.79010 

3.76411 

,.80719  i  4.18652 

.82435     4.69304 

53 

54     .77335 

3.41206 

.79038 

3.77057 

.80748  1  4.19421 

.82463 

4.70234 

54 

5o!   .77363 

3.41759 

.79067 

3.77705 

.80776 

4.20193 

.82492 

4.71166 

."«,") 

56 

.77392 

3.42312 

.79095     3.78355 

.80805 

4.20966 

.82521 

4.72102 

56 

57 

.77420 

3.42867 

.79123 

3.79007 

.80833 

4.21742 

.82549 

4.73041 

67. 

58 

.77448 

3.43424 

.79152 

3.79661 

.80862 

4.22521 

.82578 

4.73983 

68 

59 

.77477 

3.43982 

.79180     3.80316 

.80891 

4.23301 

.82607 

4.74929 

59 

80 

.77505 

3.44541  1 

.79209      3.8097'3 

.80919 

4.24084 

.82635 

4.75877' 

60 

489 


TABLE   XXIX.-NATURAL  VERSED    SIXES   AND  EXTERNAL   SECANTS. 


80° 

81° 

82°          83° 

' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec.   Vers. 

Ex.  sec. 

Vers.  Ex.  sec. 

0 

.82635 

4.75877 

.84357 

5.39245  .1  .86083   6.18530 

.87813  1  7.20551  !  0 

1 

.82664 

4.76829 

.84385 

5.40422  ;i  .86112 

6.20020 

.87842 

7.22500 

1 

.82892 

4.77784 

.84414 

5.41602  :|  .86140 

6.21517  i 

.87871 

7.24457 

2 

3 

.82721 

4.78742 

.84443 

5.42787  ii  .86169 

6.23019  ! 

.87900 

7.26425 

3 

4 

.82750 

4.79703 

.84471 

5.43977  i  -86198 

6.24529 

.87929 

7.28402 

i 

5 

.82778 

4.80667 

.84500 

5.45171 

.86227 

6.26044 

.87957 

7.30388 

5 

.82807 

4.81635 

.84529 

5.46369 

.86256 

6.27566 

.87986 

7.32384 

6 

J 

.82836 

4.82606 

.84558 

5.47572 

.86284 

6.29095 

.88015 

7.34390 

7 

8 

.82864 

4.83581 

.84586 

5.48779  i  .86313 

6.30630 

.88044 

7.36405 

8 

9  .82893 

4.84558 

.84615 

5.49991   .86:342 

6.32171 

.88073 

7.38431 

g 

10 

.82922 

4.85539 

.84644 

5.51208  !;  .86371 

6.33719 

.88102 

7.404U6 

to 

11 

.82950 

4.86524 

.84673 

5.52429   .86400 

6.35274 

.88131 

7.42511 

n 

12  .82979 

4.87511 

.84701 

5.53655  ;  .86428 

6.36835 

.88160 

7.44566 

12 

13  .83003 

4.88502 

.84730 

5.54886  : 

.86457   6.38403  , 

.88188  7.46632  i  13 

14 

.83036 

4.89497 

.84759 

5.56121 

.86486 

6.39978  \ 

.88217 

7.48707  14 

15 

.83065 

4.90495  !  .84788 

5.57361  !  .86515 

6.41560  I 

.88246 

7.50793  15 

16  |  .83094 

4.91496  ;  .84816 

5.58606 

.86544 

6.43148  ! 

.88275 

7.52S89  16 

17 

.83122 

4.92501  i  .84845 

5.59855  i 

.86573 

6.44743 

.88304 

7.54996  17 

18 

.83151 

4.93509 

.84874 

5.61110 

.86601 

6.46346 

.88333 

7.57113  18 

1!) 

.83180 

4.94521 

.84903 

5.62369  :!  .86630 

6.47955 

.88362 

7.59241  ;19 

20 

.83208 

4.95536 

.84931 

5.63633  ;i  .86659 

6.49571 

.88391 

7.61379  ;20 

21 

.83237 

4.96555 

.84960 

5.64902  !  .86688 

6.51194  !  .88420 

7.6a528  21 

32 

.83266 

4.97577 

.84989 

5.66176  !  .86717 

6.52825  Si  .88448 

7.65688  ,22 

23 

.83294 

4.98603 

.85018 

5.67454  |i  .86746 

6.54462  : 

.88477 

7.67859  23 

24 

.83323 

4.99633 

.85046 

5.68738 

1  .86774 

6.56107 

.88506 

7.70041  24 

25 

.83352 

5.00666 

.85075 

5.70027 

:  .86803 

6.57759 

.88535 

7.72234  25 

86 

.83380 

5.01703 

.85104 

5.71321  !  .86832 

6.59418  ': 

.88564 

7.74438  26 

.83409 

5.02743 

.85133 

5.72620 

.86861 

6.61085 

.88593 

7.76653  27 

28 

.83438 

5.03787  ! 

.85162 

5.73924 

.86890  6.62759 

.88622 

7.78880  28 

29 

.83467 

5.04834  i 

.85190 

5.75233 

.86919  6.64441 

.88651 

7.81118  29 

30 

.83495 

5.05886 

.85219 

5.76547 

.86947 

6.66130 

.88680 

7.83367 

30 

31 

.83524 

5.06941 

.85248 

5.77866 

.86976 

6.67826 

.88709 

7.85628 

3! 

3v> 

.83553 

5.08000  1  .85277 

5.79191  j  .87005  |  6.69530 

.88737  7.87901  32 

33 

.83581  I  5.09062  il  .85305 

5.80521  i  .87034 

6.71242 

.88766 

7.90186 

83 

84 

83610 

5.10129 

.85334 

5.81856  ;  .87063 

6.72962 

.88795 

7.92482 

31 

35 

.83639 

5.11199  1 

.85363 

5.83196  I  .87092 

6.74689 

.88824 

7.94791  135 

86 

.83667 

5.12273 

.85392 

5.84542  !  .87120 

6.76424   .88853 

7.97111  |36 

37 

.83696 

5.13350 

.85420 

5.85893  •  .87149 

6.78167   .88882 

7.99444 

87 

88 

83?'25 

5.14432 

.85449 

5.87250 

.87178 

6.79918 

.88911 

8.01788 

38 

39 

.83754 

5.15517  i 

.85478 

5.88612 

.87207 

6.81677 

.88940 

8.04146 

89 

40 

.83782 

5.16607  | 

.85507 

5.89979 

.87236 

6.83443 

.88969 

8.06515 

40 

41 

83811 

5  17700  ! 

.85536 

5.91352 

.87265 

6.85218 

.88998 

8.08897 

11 

42  i  !  83840 

5.18797  'I  .85564 

5.92731 

i  .87294 

6.87001 

.89027 

8.11292 

42 

43!  .83368 

5.19898  |!  .85593 

5.94115 

.87322 

6.88792 

.89055 

8.13699 

13 

44 

.83897 

5.21004   .856.22 

5.95505 

.87*51 

6.90592 

.89084 

8.16120 

44 

45 

83926 

5  22113   .85651 

5.96900 

.87380 

6.92400 

.89113 

8.18553 

45 

46 

.83954 

5.23226   .85680 

5.98301 

.87409 

6.94216 

.89142 

8.20999 

46 

47!  83933 

5  24343  :  .85708 

5.99708 

.87438  6.96040 

!  .89171 

8.23459 

47 

48  .84012 

5.25464 

.85737 

6.01120 

.87467  6.97873 

.89200 

8.25931  48 

49   84041 

5.26590 

.85766 

6.02538 

.87496 

6.99714 

.89229 

8.28417  49 

50  '  .84069 

5.27719 

1  .85795 

6.03962 

.87524 

7.01565 

.89258 

8.30917  J50 

51 

.84098 

5.28853 

;  .85823 

C.  05392 

1  .87553 

7.03423 

.89287 

8.33430  51 

84127 

5  29991 

.85852 

6.06828 

.87582 

7.05291 

.89316 

8.35957  52 

53 
54 

.84155 
.84184 

5.81133 

5.32279 

.85881 
.85910 

6.08269 
6.09717 

.87611 
.87640 

7.07167 
7.09052  , 

.89345 
.89374 

8.38497  53 
8.41052  54 

55 

50 
57 

58 

.84213 
.84242 
.84270 
.84299 

5.33429 
5.34584 
5.35743 
5.36906 

.85939 
.85967 
.85996 
.86025 

6.11171 
6.12630 
6.14096 
6.15568 

.87669 
.87698 
.87726 
.87755 

7.10946 
7.12849 
7.14760 
7.16681 

.89403 
.89431 
.89460 
.89489 

8.43620  i55 
8.46203  56 
8.48800  157 
8.51411  58 

59 

00 

.84328 
.84357 

5.38073 
5.39245 

.86054 

.86083 

6.17046 
6.18530 

.87784 
!  .87813 

7.18612 
7.20551  1 

.89518 
1  .89547 

8.54037  |59 
8.56677  !60 

490 


TABLE    XXIX.— NATURAL  VERSED  SINES  AND  EXTERNAL    SECANTS. 


J 

|4o 

* 

J5° 

£ 

6° 

/ 

Vers. 

Ex.  sec. 

Yers. 

Ex.  sec. 

Yers. 

Ex.  sec. 

0 

.89547 

8.56677 

.91234 

10.47371 

.93024 

13.33559 

0 

1 

.89576 

8.59332 

.91313 

10.51199 

.93053 

13.39547 

1 

2 

.89605 

8.62002 

.91342 

10.55052 

.93082 

13.45586 

2 

3 

.89034 

8.64687 

.91871 

10.58932 

.93111 

13.51676 

3 

4 

.89663 

8.67387 

.91400 

10.62837 

.93140 

13.57817 

4 

5 

.89892 

8.70103 

.91429 

10.66769 

.93169 

13.64011 

5 

6 

.89721 

8.72333 

.91458 

10.70728 

.93198 

13.70258 

6 

7 

'.89750 

8.75579 

.91487 

10.74714 

.93227 

13.76558 

7 

8 

.89779 

8.78341 

.91516 

10.78727 

.93257 

13.82913 

8 

9 

.89808 

8.81119 

.91545 

10.82768 

.93286 

13.89323 

9 

10 

.89836 

8.83912 

.91574 

10.86837 

.93315 

13.95788 

10 

11 

.89865 

8.86722 

.91603 

10.90934 

.93344 

14.02310 

11 

12 

.89894 

8.89547 

.91032 

10.95060 

.93373 

14.08890 

12 

13 

.89923 

8.92389 

.91061 

10.99214 

.9:3402 

14.15527 

13 

14 

.89952 

8.95248 

.91690 

11.03397 

.93431 

14.22223 

14 

15 

.89931 

8.98123 

.91719 

11.07610 

.93460 

14.28979 

15 

16 

.90010 

9.01015 

.91748 

11.11852 

.93489 

14.35795 

16 

17 

.90J39 

9.03923 

.91777 

11.16125 

.93518 

14.42672 

17 

18 

.90038 

9.08349 

.91806 

11.20127 

.93547 

14.49611 

18 

19 

.90097 

0.09792 

.91835 

11.24761 

.93576 

14.56614 

19 

20 

.9C126 

9.12752 

.91864 

11.29125 

.93605 

14.63679 

20 

21 

.90155 

9.15730 

.91893 

11.33521 

.93634 

14.70810 

21 

22 

.90184 

9.18725 

.91922 

11.37948 

.93663 

14.78005 

22 

23 

.90213 

9.21739 

!  .91951 

11.42408 

.93692 

14.85268 

23 

24 

.90242 

9.24770 

i  .91980 

11.46900 

.93721 

14.92597 

24 

25 

.90271 

9.27819 

.92009 

11.51424 

.93750 

14.99995 

25 

20 

.90300 

9.30887 

.92038 

11.55982 

.93779 

15.07462 

26 

27 

.'JO;-.'!) 

9.33973 

.92067 

11.60572 

.93803  ' 

15.14999 

27 

28 

.90338 

9.37077 

.92096 

11.65197 

.93837 

15.22607 

28 

29 

.90-^86 

9.40201 

.92125 

11.69356 

.93866 

15.30237 

29 

30 

.90415 

9.43343 

.92154 

11.74550 

.93895 

15.38041 

30 

31 

.90444 

9.46505 

.92183 

11.79073 

.93924 

15.45869 

31 

32 

.90473 

9.49685 

.92212 

11.84042 

.93953 

15.53772 

32 

33 

.90502 

9.52886 

.92241 

11.88841 

.93982 

15.61751 

33 

34 

.90531 

9.56106 

.92270 

11.93677 

.94011 

15.69808 

34 

35 

.90560 

9.59346 

.92299 

11.98549  i 

.94040 

15.77944 

35 

36 

.90589 

9.62605 

.92328 

12.03450  1 

.94069 

15.86159 

36 

37 

.90618 

9.65885 

.92357 

12.08040  ; 

.94098 

15.94456 

37 

38 

.90647 

9.69186 

.92386 

12.13388 

.94127 

16.02835 

38 

39 

.90676 

9.72507 

.92415 

12.18411 

.94156 

16.11297 

39 

40 

.90705 

9.75849 

.92444 

12.23472 

.94186 

16.19843 

40 

41 

.90734 

9.79212 

.92473 

12.2857-3 

.94215 

16.28476 

41 

42 

.90763 

9.82596 

.92502 

12.33712 

.94244 

16.37196 

42 

43 

.90792 

9.86001 

.92531 

12.38891 

.94273 

16.46005 

43 

44 

.90821 

9.89428 

.92560 

12.44112 

.94302 

16.54903 

44 

45 

.90350 

9.92877 

.92589 

12.49373 

.94331 

16.63893 

45 

46 

.90879 

9.96348 

.92618 

12.54676 

.94360 

16.72975 

46 

47 

.90908 

9.99841 

.92647 

12.60021 

.94389 

16.82152 

47 

48 

.90937 

10.03356 

.92676 

12.65408 

.94418 

16.91424 

48 

49 

.90966 

10.06894 

.92705 

12.70838 

.94447 

17.00794 

49 

50 

.90995 

10.10455 

.92734 

12.76312 

.94476 

17.10262 

50 

51 

.91024 

10.14039 

.92763 

12.81829 

.94505 

17.19830 

51 

5-3 

.91053 

10.17646 

.92792 

12.87391 

.94534 

17.29501 

52 

63 

.91082 

10.21277 

.92821 

12.92999 

.94563 

17.39274 

53 

54 

.91111 

10.24932 

.92850 

12.98651 

.94592 

17.49153 

54 

55 

.91140 

10.28610 

.92879 

13.04350 

.94621 

17.59139 

55 

56 

.91169 

10.32313 

.92908 

13.10096 

.94650 

17.69233 

56 

57 

.91197 

10.36040 

.92937 

13.15889 

.94679 

17.79438 

57 

58 

.91226 

10.39792 

.92966 

13.21730 

.94708 

17.89755 

58 

59 

.91255 

10.43569 

.92995 

13.27'620 

.94737 

18.00185 

5,0 

60 

.91284 

10.47371 

.93024 

13.33559 

.94766 

18.10732 

60 

491 


TABLE  XXIX.— NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS. 


/ 

{ 

S7° 

8 

8° 

B 

9° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.94766 

18.10732 

.96510 

27.65371 

.98255 

56.29869 

0 

1 

.94795 

18.21397 

.96539 

27.89440 

.98284 

57.26976 

1 

2 

.94825 

18.32182 

.96568 

28.13917 

.98313 

58.27431 

2 

3 

.94854 

18.43088 

.96597 

28.38812  i 

.98342 

59.31411 

3 

4 

.94883 

18.54119 

.96626 

28.64137 

.98371 

60.39105 

4 

5 

.94912 

18.65275 

.96655 

28.89903 

.98400 

61.50715 

5 

6 

.94941 

18.76560 

.96684 

29.16120  ! 

.98429 

62.66460 

6 

7 

.94970 

18.87976 

.96714 

29.42802 

.98458 

63.86572 

7 

0 

.94999 

18.99524 

.96743 

29.69960 

.98487 

65.11304 

8 

9 

.95028 

19.11208 

.90772 

29.97607 

.98517 

66.40927 

9 

10 

.95057 

19.23028 

.96801 

30.25758 

.98546 

67.75736 

10 

11 

.95086 

19.34989 

.96830 

30.54425 

.98575 

G9.  16047 

11 

12 

.95115 

19.47093 

.96859 

30.83623 

.98604 

70.62285 

12 

13 

.95144 

19.59341 

.96888 

31.13366 

.98633 

72.14583 

13 

11 

.95173 

19.71737 

.96917 

31.43671 

.98662 

73.7'3580 

14 

15 

.95202 

19.84283 

.96946 

31.74554 

.98691 

75.39655 

15 

10 

.95231 

19.96982 

.96975 

32.06030 

.98720 

77.13274 

16 

17 

.95280 

20.09838 

.97004 

32.38118 

.98749 

78.94968 

17 

18 

.95289 

20.22852 

.97033 

32.70835 

.98778 

80.85315 

18 

19 

.95318 

20.36027 

.97062 

33.04199 

.98807 

82.84947 

19 

20 

.95347 

20.49368 

.97092 

33.38232 

.98836 

84.94501 

20, 

21 

.95377 

20.62876 

.97121 

33.72952 

.88866 

87.14924 

21 

22 

.95406 

20.76555 

.97150 

34.08380 

.98895 

89.4G8SO 

22 

23 

.95435 

20.90403 

.97179 

34.44539 

.98924 

91.91387 

23 

24 

.954(54 

21.04440 

.97208 

34.81452 

.98953 

94.49471 

24 

25 

.95-193 

21.18653 

.97237 

35.19141 

.98982 

97.22303 

25 

20 

.95522 

21.33050 

.97206 

35.57633 

.C9011 

100.1119 

26 

27 

.95551 

21.47635 

.97295 

35.96953 

.99040 

103.1757 

27 

28 

.95580 

21.62413 

.97324 

36.37127 

.99069 

106.4311 

28 

29 

.95609 

21  .  77386 

.97a53 

36.78185 

.99098 

109.8906 

29 

30 

.95638 

21.92559 

.97382 

37.20155 

.99127 

113.5930 

30 

31 

.95667 

22.07935 

.97411 

37.63068 

.£9156 

117.5444 

31 

32 

.95696 

22.23520 

.97440 

38.06957 

.99186 

121.7780 

32 

33 

.95725 

22.39316 

.97470 

38.51855 

.99215 

126.3253 

33 

34 

.95754 

22.55329 

.97499 

38.977(J7 

.99244 

131.2223 

34 

35 

.95783 

22.71563 

.97523 

39.44820 

.99273 

136.5111 

35 

36 

.95812 

22.88022  ' 

.97557 

39.92903 

.99302 

142.2406 

36 

37 

.95842 

23.04712 

.97'586 

40.42206 

.99331 

148.4084 

37 

38 

.95871 

23.21637 

.97615 

40.92772 

.99860 

155.2623 

38 

39 

.95900 

23.38802 

.97644 

41.44525 

.99-389 

162.7033 

39 

40 

.95929 

23.56212 

.97673 

41.97571 

.S9418 

170.8883 

40 

41 

.95958 

23.73873 

.97702 

42.51961 

.99447 

17'9.9350 

41 

42 

.95987 

23.91790 

.97731 

43.07745 

.9947'6 

189.9808 

42 

43 

.96016 

24.09969 

.97760 

43.64980 

.99505 

201.2212 

43 

44 

.96045 

24.28414 

.97789 

44.23720 

.995&5 

213.8600 

44 

45 

.96074 

24.47134 

.97819 

44.84026 

.99564 

228.18:39 

45 

46 

.96103 

24.66132 

.97848 

45.45963 

.99593 

244.5540 

46 

47 

.96132 

24.85417 

.97877 

46.09596 

.99622 

263.4427 

48 

.96161 

25.04994 

.97906 

46.74997 

.99651 

285.4795 

48 

49 

.96190 

25.24869 

.97935 

47.42241 

.99080 

311.5230 

49 

50 

.96219 

25.45051 

.97964 

48.11406 

.99709 

342.7752 

50 

51 

.96248 

25.65546 

;  .97993 

48.82576 

.99738 

380.9723 

51 

53 

.96277 

25.86300 

.98022 

49.55840 

.99707 

428.7187 

52 

53 

.96307 

26.07503 

.98051 

50.31290 

.99796 

490.1070 

53 

54 

.96336 

26.28981 

98080 

51.09027  ! 

.99825 

571.9581 

54 

55 

.96365 

26.50804 

!98109 

51.89150 

.99855 

686.5490 

55 

56 

.96394 

26.72978 

.98138 

52.71790 

.99884 

858.4309 

56 

57 

.96423 

26.95513 

.98168 

53.57046 

.99913 

1144.910 

57 

58 

.96452 

27.18417 

.98197 

54.450.-.:} 

.99942 

1717.874 

58 

59 

.96481 

27.41700 

.98226 

55..  35946 

.99971 

3436.747 

59 

60 

.96510 

27.65371 

.98255 

56.29869 

1.00000 

Infinite 

60 

TABLE  XXX.-CUBIC  YARDS  PER  100  FEET.      SLOPES 


Depth 

Base 

Base 

|  Base 

I  Base 

Base 

Base 

Base 

Base 

d 

12 

14 

16 

|  18 

22 

24 

26 

28 

1 

45 

53 

60 

68 

82 

90 

97 

105 

2 

93 

107 

122 

137 

167 

181 

196 

211 

3 

142 

163 

186 

208 

253 

275 

297 

319 

4 

193 

222 

252 

281 

341 

370 

400 

430 

5 

245 

282 

319 

356 

431 

468 

505 

542 

6 

300 

344 

389 

433 

522 

567 

611 

656 

356 

408 

460 

612 

616 

668 

719 

771 

8 

415 

474 

533 

593 

711 

770 

830 

889 

9 

475 

542 

608 

675 

808 

875 

942 

1008 

10 

537 

611 

685 

759 

9or 

981 

1056 

1130 

11 

601 

682 

764 

845 

1008 

1090 

1171 

1253 

12 

667 

756 

844 

933 

1111 

1200 

1289 

1378 

13 

734 

831 

926 

1023 

1216 

1312 

1408 

1505 

14 

804 

907 

1010 

1115 

1322 

1426 

1530 

1633 

15 

875 

986 

1096 

1208 

1431 

1542 

1653 

1764 

16 

948 

1067 

1184 

1304 

1541 

1659 

1778 

1896 

1  7 

1023 

1149 

1274 

1401 

1653 

1779 

1905 

2031. 

18 

1100 

1233 

1366 

1500 

1767 

1900 

2033 

2167 

19 

1179 

1319 

1460 

1601 

1882 

2023 

2164 

2305 

2> 

1259 

1407 

1555 

1704 

2000 

2148 

2296 

2444 

21 

1342 

1497 

1653 

1808 

2119 

2275 

2431 

2586 

K 

1426 

1589 

1752 

1915 

2241 

2404 

2567 

2730 

23 

1512 

1682 

ia53 

2023 

2364 

2534 

2705 

2875 

2i 

1600 

1778 

1955 

2133 

2489 

2667 

2844 

3022 

25 

1690 

1875 

2060 

2245 

2616 

2801 

2986 

3171 

26 

1781 

1974 

2166 

2359 

2744 

2937 

3130 

-  3322 

1875 

2075 

2274 

2475 

2875 

3075 

3275 

3475 

23 

1970 

2178 

2384 

2593 

3007 

3215 

3422 

3630 

29 

2068 

2282 

2496 

2712 

3142 

3356 

3571 

3786 

33 

2167 

2389 

2610 

2833 

3278 

3500 

3722 

3944 

31 

2233 

2497 

2726 

2956 

3416 

3645 

3875 

4105 

32 

2370 

2607 

2844 

3081 

3556 

3793 

4030 

4267 

33 

2475 

2719 

2964 

3208 

3697 

3942 

4186 

4431 

34 

2581 

2833 

3085 

3337 

3841 

4093 

4344 

4596 

35 

2690 

2949 

3208 

3468 

3986 

4245 

4505 

4764 

36 

2800 

3067 

3333 

3600 

4133 

4400 

4667 

4933 

37 

2912 

3186 

3460 

3734 

4282 

4556 

4831 

5105 

38 

3026 

3307 

3589 

3870 

4433 

4715 

4996 

5278 

39 

3142 

3431 

3719 

4008 

4586 

4875 

5164 

5453 

40 

3259 

3550 

3852 

4148 

4741 

5037 

5333 

5630 

41 

3379 

3682 

3986 

4290 

4897 

5201 

5505 

5F08 

4-2 

3500 

3811 

4122 

4433 

5056 

5367 

5678 

5989 

43 

3623 

3942 

4260 

4579 

5216 

5534 

5853 

6171 

44 

3748 

4074 

4400 

4726 

5378 

5704 

6030 

6356 

45 

3875 

4208 

4541 

4875 

5542 

5875. 

6208 

6542 

40 

4004 

4344 

4684 

5026 

5707 

6048 

6389 

6730 

47 

4134 

4482 

4830 

5179 

5875 

6223 

6571 

6919 

48 

4267 

4622 

4978 

5333 

6044 

6400 

6756 

7111 

49 

4401 

4764 

5127 

5490 

6216 

6579 

6942 

7305 

50 

4537 

4907 

5278 

5648 

6389  ; 

6759 

7130 

7500 

51 

4675 

5053 

5430 

5808 

6564 

6942 

7319 

7697 

52 

4815 

5200 

5584 

5970 

6741 

7126 

7511 

7S96 

53 

4950 

5349 

5741 

6134 

6919 

7312 

7705 

8097 

54 

5100 

5500 

5900 

6300 

7100 

7500 

7900 

8300 

55 

5245 

5653 

6060 

6468 

7282 

7690 

8097 

8505 

56 

5393 

5807 

6222 

6637 

74(57 

7881 

8296 

8711 

57 

5542 

5964 

6386 

6808 

7653 

8075 

8497 

8919 

58 

5693 

6122 

6552 

6981 

7841 

8270 

8700 

9130 

59 

5845 

6282 

6719 

7156 

8031 

8468 

8905 

9342 

60 

6000 

6444 

6889 

7333 

8222 

8667 

9111 

9556 

i 

493 


TABLE  XXX.— CUBIC  YARDS   PER  100  FEET.      SLOPES  ^  :  1. 


Depth 
d 

Base 
12 

Base  !  Base 
14    16 

Base 
18 

Base 
22 

Base   Base 
24    26 

Base 
28 

1 

46 

54 

Gl     69 

83     91 

98 

106 

2 

96 

111 

126  j   141 

17'0 

185 

200 

215 

3 

150 

172 

194 

217 

261 

283 

306 

328   • 

4 

207 

237 

267 

296 

356 

385 

415 

444 

5 

269 

306 

343 

380 

454 

491 

528 

565 

6 

333 

378 

422 

467 

556 

600 

644 

689 

402 

454 

506 

557 

661 

713 

765 

817 

8 

474 

533 

593 

652 

770 

830 

889 

948 

9 

550 

617 

683 

750 

883 

950 

1017 

1083 

10 

630 

704 

778 

852 

1000 

1074 

1148 

1222 

11 

713 

794 

876 

957 

1120 

1202 

1283 

1365 

12 

800 

889 

978 

1067 

1244 

1333 

1422 

1511 

13 

891 

987 

1083 

1180 

1372 

1469 

1565 

1661 

14 

985 

1089 

1193 

1296 

1504 

1607 

1711 

1815 

15 

1083 

1194 

1306 

1417 

1639 

1750 

18G1 

1972 

16 

1185 

1304 

1422 

1541 

1779 

1896 

2015 

2133 

%  17 

1291 

1417 

1543 

1669 

1920 

2046 

2172  !  2298 

18 

1400 

1533 

1667 

1800 

20G7 

2200 

2333 

2467 

19 

1513 

1654 

1794 

1935 

2217 

2357 

2498 

2639 

20 

1630 

1778 

1926 

2074 

2370 

2519 

2667 

2815 

21 

1750 

1906 

2061 

2217 

2528 

2683 

2839 

2994 

22 

1874 

2037 

2200 

2363 

2689 

2852 

3015 

3178 

23 

2002 

2172 

2343 

2513 

2854 

3024 

3194 

3365 

24 

2133 

2311 

2489 

2667 

3022 

3200 

3378 

3556 

25 

2269 

2454 

2639 

2824 

3194 

3380 

3565 

87oO 

26- 

2407 

2600 

2793 

2085 

3370 

35C3 

3756 

3948 

27 

2550 

2750 

2950 

3150 

3550 

3750 

3950 

4151 

28 

2696 

2904 

3111 

3319 

3733 

3941 

4148 

4356 

29 

2846 

3061 

3276 

3491 

3920 

4135  !  4350 

4565 

30 

3000 

3222 

3444 

3667 

4111 

4333 

4556 

4778 

31 

3157 

3387 

3617 

3846 

4306 

4535 

4765 

4994 

32 

3319 

3556 

3793 

4030 

4504 

4741 

4978 

5215 

33 

3483 

3728 

3972 

4217 

4706 

4950 

5194 

5439 

34 

3652 

3904 

4156 

4407 

4911 

5163 

5415 

6667 

35 

3824 

4083 

4343 

4602 

5120 

5380 

5639 

5898 

36 

4000 

4267 

4533 

4800 

5333 

5600 

5867 

6133 

37 

4180 

4454 

4728 

5002 

5550    5824 

6098 

6372 

38 

4363 

4644 

4926 

5207 

5770  1  G052 

6333 

6615 

39 

4550 

4839 

5128 

5417 

5994 

6283 

6572 

6861 

40 

4741 

5037 

5333 

5630 

6222 

6519 

6815 

7111 

41 

4935 

5239 

5543 

5846 

6454 

6757 

7061 

7365 

42 

5133 

5444 

5756 

6067 

6689 

7000 

7311 

7623 

43 

5335 

5654 

5972 

6291 

6928 

7246 

7565 

7883 

44 

5541 

5867 

6193 

6519 

7170 

7496 

7822 

8148 

45 

5750 

6083 

6417 

6750 

7417 

7750 

8083 

8417 

46 

5963 

6304 

6644 

6985 

7667 

8007 

8348 

8G89 

47 

6180 

6528 

6876 

7224 

7920 

8269 

8617 

89G5 

48 
49 

6400 
6624 

6756 
6987 

7111 
7350 

7467 
7713 

8178 
8439 

8802 

m 

9244 
9528 

50 

6852 

7222 

7593 

7963 

87C4 

9074 

9444 

9815 

51 

7083 

7461 

7839 

8217 

8972 

9350 

9728 

10106 

52 

7319 

7704 

8089 

8474 

9244 

9630 

10015 

10400 

53 

7557 

7950 

8343 

8735 

9520 

9913 

10306 

10G98 

54 

7800 

8200 

8600 

9000 

9800 

10200 

10GOO 

11000 

55 

8046 

8454 

8861 

92G9 

10083 

10491 

10898 

11806 

56 

8296 

8711 

9126 

9541 

10370 

10785 

11200 

11615 

57 

8550 

8972 

9394 

9817 

10661 

11083 

11506 

11928 

58 

8807 

9237 

9667 

10096 

10956 

11385 

11815  i  12244 

59 

9069 

9506    9943 

10380 

11254 

11691 

12128   12565 

60 

9333 

9778  !  10222 

10667 

11556 

12000 

12444   12889 

' 

494 


TABLE  XXX.— CUBIC  YARDS  PER  100  FEET.      SLOPES  1  :  1. 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

d 

12 

14 

16 

18 

20 

23 

30 

32 

1 

48 

56 

63 

70 

78 

107 

115 

122 

2 

104 

119 

183 

148' 

163 

222 

237 

252 

3 

167 

189 

211 

233  ' 

256 

344 

367 

389 

4 

237 

267 

296 

326 

356 

474 

504 

533 

5 

315 

352 

389 

426 

463 

611 

648 

685 

6 

400 

444 

489 

533 

578 

756 

800 

844 

493 

544 

596 

648 

700 

907 

959 

1011 

8 

593 

652 

711 

770 

830 

1067 

1126 

1185 

9 

700 

767 

833 

900 

967 

1233 

1300 

1367 

10 

815 

889 

963 

1037 

1111 

1407 

1481 

1556 

11 

937 

1019 

1100 

1181 

1263 

1589 

1670 

1752 

12 

1067 

1156 

1244 

13*3 

1422 

1778 

1867 

1956 

13 

1204 

1300 

1396 

1493 

1589 

1974 

2070 

2167 

14 

1348 

1452 

1556 

1659 

1763 

2178 

2281 

2385 

15 

1500 

1611 

1722 

1833 

1944 

2389 

2500 

2611 

16 

1659 

1778 

1896 

2015 

2133 

2607 

2726 

2844 

17 

1826 

1952 

2078 

2204 

2330 

2833 

2959 

3085 

18 

2000 

2133 

2267 

2400 

2533 

3067 

3200 

3333 

19 

2181 

2322 

2*3 

2604 

2744 

3307 

3448 

3589 

20 

2370 

2519 

2667 

2815 

2963 

3556 

3704 

3852 

21 

2567 

2722 

2878 

3033 

3189 

3811 

3967 

4122 

22 

2770 

2933 

3096 

3259 

3422 

4074 

4237 

4444 

23 

2981 

3152 

3322 

3493 

3663 

4344 

4515 

4685 

24 

3200 

3378 

3556 

3733 

3911 

4622 

4800 

4978 

25 

3426 

3611 

3796 

3981 

4167 

4907 

5093 

5278 

26 

3659 

3852 

4044 

4237 

4430 

5200 

5393 

5585 

2?' 

3900 

4100 

4300 

4500 

4700 

5500 

5700 

5900 

28 

4148 

4356 

4563 

4770 

4978 

5807 

6015 

6222 

29 

4404 

4619 

4833 

5048 

5263 

6122 

6337 

6552 

30 

4667 

4889 

5111 

5333 

5556 

6444 

6667 

6889 

31 

4937 

5167 

5396 

5626 

5856 

6774 

7004 

7233 

32 

5215 

5452 

5689 

5926 

6163 

7111 

7348 

7585 

33 

5500 

5744 

5989 

6233 

6478 

7456 

7700 

7944 

84 

5793 

6044 

6296 

6548 

6800 

7807 

8059 

8311 

35 

6093 

6352 

6611 

6870 

7130 

8167 

8426 

8685 

36 

6400 

6667 

6933 

7200 

7467 

8533 

8800 

9067 

37 

6715 

6989 

7263 

7537 

7811 

8907 

9181 

9456 

33 

7037 

7319 

7600 

7881 

8163 

9289 

9570 

9852 

39 

7367 

7656 

7944 

8233 

8522 

9678 

9967 

10256 

40 

7704 

8000 

8296 

8593 

8889 

10074 

10370 

10667 

41 

8048 

8352 

8656 

8959 

9263 

10478 

10781 

11085 

42 

8400 

8711 

9022 

9333 

9644 

10889 

11200 

11511 

43 

8759 

9078 

9396 

9715 

10033 

11307 

11626 

11944 

44 

9126 

9452 

9778 

10104 

10430 

11733 

12059 

12385 

45 

9500 

9833 

10167 

10500 

10833 

12167 

•12500 

12833 

46 

9881 

10222 

10563 

10904 

11244 

12607 

12948 

13289 

47 

10270 

10619 

10967 

11315 

11663 

13056 

13404 

13752 

48 

10667 

11022 

11378 

11733 

12089 

13511 

13867 

14222 

49 

11070 

11433 

11796 

12159 

12522 

13974 

14337 

14700 

50 

11481 

11852 

12222 

12593 

12963 

14444 

14815 

15185 

51 

11900 

12278 

12656 

13033 

13411 

14922 

15300 

15678 

52 

12326 

12711 

13096 

13481 

13867 

15407 

15793 

16178 

53 

12759 

13152 

13544 

13937 

14330 

15900 

16293 

1G685 

54 

13200 

13600 

14000 

14400 

14800 

16400 

16800 

17200 

55 

13648 

14056 

14463 

14870 

15278 

16907 

17315 

17722 

56 

14104 

14519 

14933 

15348 

15763 

17422 

17837 

18252 

57 

14567 

14989 

15411 

15833 

16256 

17944 

18367 

18789 

58 

15037 

15467 

15896 

16326 

16756 

18474 

18904 

19333 

59 

15515 

15952 

16389 

16826 

17263 

19011 

19448 

19885 

60 

16000 

16444 

16889 

17333 

17778 

19556 

20000 

20444 

495 


TABLE  XXX.— CUBIC  YARDS  PER  103  FEET.      SLOPES  \y>  :  1. 


Depth 
d 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

50 

57 

65 

73 

80 

109 

117 

124 

2 

111 

126 

141- 

158 

170 

230 

244 

259 

3 

183 

206 

228 

250 

272 

361     383 

406 

4 

267 

296 

326 

356 

385 

504    533 

563 

5 

361 

398 

435 

47'2 

509 

037     694 

731 

6 

4G7 

511 

556    GOO 

644 

822    867 

911 

7 

583 

635 

687    739 

791 

998  J  1050 

1102 

8 

711 

770 

830 

889 

948 

1185  !  1244 

1304 

9 

850 

917 

983 

1050 

1116 

1383 

1450 

1517 

10 

1000 

1074 

1148 

1222 

1296 

1593 

1667 

1741 

11 

1161 

1243 

1324 

1406 

1487 

1813 

1894 

1976 

12 

1333 

1422 

1511 

1600 

1689 

2044 

2133 

2222 

13 

1517 

1613 

1709 

1806 

1902 

2287 

2383 

2480 

14 

1711 

1815 

1919 

2022 

2126 

2541 

»•  i  i 

27-48 

15 

31)17 

2028 

2139 

2250 

2361 

2806 

2917 

3028 

16 

2133 

2252 

2370 

2489 

2607 

3081 

3200 

3319 

17 

2301 

2487 

2613 

2739 

2865 

3369 

3494 

3620 

18 

2600 

2733 

2867 

3000 

was 

3667 

3800 

3933 

19 

2850 

2991 

3131 

3272 

3413 

3976 

4117 

4257 

.  2.0 

3111 

3259 

3407 

3556 

3704 

4296 

4444 

4592 

21 

3383 

3539 

3694 

3850 

4005 

4628 

4783 

4939 

22 

3667 

3830 

3993 

4156 

4318 

4970 

5133    5296 

23 

3961 

4131 

4302 

4472 

4642 

5324 

5494  '  5665 

24 

4267 

4444 

4622 

4800 

4078 

5689 

58G7 

6044 

25 

4583 

4769 

4954 

5139 

5324 

6065 

6250 

6435 

26 

4911 

5104 

5296 

5489 

5681 

6452 

66-44 

6837 

27 

5250 

5450 

5650 

5850 

6050 

6850 

7050 

7250 

28 

5600 

5807 

6015 

6222 

6430 

7259 

7467 

7674 

29 

5961 

6176 

6391 

»  6606 

6820 

7680 

7894 

8109 

30 

6333 

6556 

6778 

7000 

7222 

8111 

8333 

8555 

31 

6717 

6946 

7176 

7406 

7635 

8554 

8783 

9013 

32 

7111 

7348 

7585 

7822    8059 

9007 

!£41 

9482 

33 

7517 

7761 

8006 

8250 

8494 

1472 

9717 

9962 

34 

7933 

8185 

8437 

8689 

8941 

9948 

10800 

10452 

35 

8361 

8620 

8880 

9139 

9398 

10435 

10694 

10954 

36 

8800 

9067 

9333 

9600 

9867 

10933 

11200 

11467 

37 

9250 

9524 

9798 

10072 

10346 

11443 

11717 

11991 

38 

9711 

9993 

10274 

10556 

10837 

11963 

12244 

1252G 

39 

10183 

1047'2  !  10761 

11050 

11339 

12494 

12788 

13072 

40 

10667 

10963   11259 

11556 

11852 

13037 

13333 

13630 

41 

11161 

11465   11769 

12072 

12376 

13591 

13894 

14198 

42 

11667 

11978  '  12289 

12600 

12911 

14156 

14467 

14778 

43 

12183 

12502  i  12820 

13139   13457   14731 

15050 

15369 

44 

12711 

13037  !  1.3363 

13689 

14015   15319 

15644 

15970 

45 

13250 

13583 

13917 

14250 

14583 

15917 

16250 

16583 

46 

13800 

14141 

14481 

14822   15163 

16526 

168G7   17207 

47 

14361 

14709 

15057 

15406   15754 

17146 

17194   17843 

48 

14933 

15289 

15644 

16000   16356 

17778 

18133   18489 

49 

15517 

15880 

16243 

16606 

16968 

18420 

1S7K3  '  1011(5 

50 

16111 

16481 

16852 

17222 

17592 

19074 

11)111    19815 

51 

16717 

17094 

17472 

17850 

18228 

19739 

20117   2045)4 

52 

173.53 

17719 

18104 

18489 

18874 

20415 

r:nsiW   21  1S5 

53 

17961 

18354 

1S7'46 

19139 

19531 

21102 

21494 

21887 

54 

18300 

19000 

19400 

10800 

20200 

21800 

22200 

22600 

55 

19250 

19657 

20065 

20472 

20880 

22509 

22917  !  S332-i 

56 

19911 

20326 

20741 

21156 

21570 

23230 

23044   240.-)'.) 

57 

20583 

21006 

21428 

21850 

2227'2 

23961 

243K-J   24805 

58 

21267 

21696 

22126 

22556 

22985 

247'04 

25133   25563 

59 

21961 

22398 

2288S5 

83272 

237'09 

85457 

2.V'!ll   2G332 

GO 

2-20(17 

23111 

23556 

24000 

24444   26222 

26G67   27111 

49G 


TABLE  XXX.— CUBIC  YARDS  PER  100  FEET.   SLOPES  2  :  1. 


Depth 
d 

Base   Bass 
12    14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

l 

56 

63 

70 

78 

85 

~15 

122 

130 

2 

133 

148 

163 

178 

193 

252 

267 

281 

3 

233 

256 

278 

300 

322 

411 

433 

456 

4 

356 

385 

415 

444 

47'4 

593 

622 

652 

5 

500 

537 

574 

611 

648 

796 

833 

87'0 

6 

667 

711 

756 

800 

844 

1022 

1067 

1111 

856 

907 

959 

1011 

1063 

1270 

1322 

1374 

8 

1067 

1126 

1185 

1244 

1304 

1541 

1600 

1659 

9 

1300 

1367 

1433 

1500 

1567 

1833 

1900 

196?' 

10 

1556 

1630 

1704 

1778 

1852 

2148 

2222 

2296 

11 

1833 

1915 

1996 

2078 

2159 

2485 

2567 

2648 

12 

2133 

2222 

2311 

2400 

2489 

2844 

2933 

3022 

m 

2456 

2552 

2648 

2744 

2841 

3226 

3322 

3419 

14 

2800 

2904 

3007 

3111 

3215 

3630 

3733 

3837 

15 

3167 

3278 

3389 

3500 

3611 

4056 

4167 

4278 

16 

3556 

3674 

3793 

3911 

4030 

4504 

4622 

4741 

17 

3967 

4093 

4219 

4344 

4470 

4974 

5100 

5226 

18 

4400 

4533 

4667 

4800 

4933 

5467 

5600 

6738 

19 

4856 

4996 

5137 

5278 

5419 

5981 

6122 

6263 

20 

5333 

5481 

5630 

5778 

5926 

6519 

6667 

6815 

21 

5833 

5989 

C144 

6300 

6456 

707'8 

7233 

7389 

••>••> 

6356 

0519 

6681 

6844 

7007 

7659 

7822 

7985 

23 

6900 

707'0 

7241 

7411 

7581 

8263 

8433 

8504 

24 

7487 

7644 

7822 

8000 

8178 

8889 

9067 

9144 

25 

8056 

8241 

8426 

8611 

8796 

9537 

rt7'22 

9807 

2C, 

8667 

8859 

9052 

9244 

9437 

10207 

10400 

10593 

27 

9300 

9500 

9700 

9900 

10100 

10900 

11100 

11200 

28 

9956  1  10163 

10370 

10578 

10785 

11615 

11822 

12030 

29 

10633 

10848 

11063 

11278 

11493 

12352 

12567 

12781 

30 

11333 

11556 

11778 

12000 

12222 

13111 

13333 

13556 

31  - 

12056 

12285 

12515 

12744 

12974 

13893 

14122 

14352 

32 

12800 

13037 

13274 

13511 

13748 

14U96 

14933 

1517'0 

S3 

13567 

13811 

14056 

14300 

14544 

15522 

15767 

16011 

34 

14356 

14607 

14859 

15111 

15363 

16370 

16622 

16874 

35' 

15167 

15426 

15685 

15944 

16204 

17241 

17500 

17759 

36 

16000 

16267 

16533 

16800 

17067 

18133 

18400 

18667 

37 

16856 

17130 

17404 

17678 

17952 

19048 

19322 

19596 

38 

17733 

18015 

18296 

18578 

18859 

19985 

20267 

20548 

39 

18633 

18922 

19211 

19500 

19789 

20944 

21233 

21522 

40 

19556 

19852 

20148 

20444 

20741 

21926 

22222 

22516 

41 

20500 

20804 

21107 

21411 

21715 

22930 

23233 

23537 

42 

21467 

21778 

22089 

22400 

22711 

23956 

24267 

24578 

43 

23456 

22774 

23093 

23411 

23730 

25004 

25322 

25641 

44 

23467 

23793 

24119 

24444 

24770 

26074 

26400 

26726 

45 

21500 

24833 

25167 

25500 

25833 

27167  • 

27500 

27833 

46 

25556 

25896 

26237 

26578 

26919 

28281 

28622 

28963 

47 

26633 

26981 

27330 

27678 

28026 

29419 

29767 

30115 

48 

27733 

28089 

28444 

28800 

29156 

30578 

30933 

31289 

49 

28856 

29219 

29581 

29944 

30307 

31759 

32122 

32485 

50 

30000  |  30370 

30741 

31111 

31481 

32963 

33333 

33704 

51 

31167 

31544 

31922 

32300 

32678 

34189 

34567 

34944 

52 

32a56 

32741 

33126 

33511 

33396 

35437 

35822 

3C207 

53 

33567 

33959 

34352 

34744 

35137 

36707 

37100 

37493 

54 

34800 

35200 

35600 

36000 

36400 

38000 

38400 

38800 

55 

36056 

36463 

36870 

3?'278 

37(185 

39315 

39722 

40130 

56 

37333 

37748 

38163 

38578 

38993 

40652 

41067 

41481 

57 

38633 

39056 

39478 

39900 

40322 

42011 

42433 

42856 

58 

39956 

40385 

40815 

41244 

41674  i  43393 

43822 

44252 

59 

41300 

41737 

42174 

42611 

43048  ]  44796 

45233 

45670 

60 

42667 

43111 

43556 

44000 

44444 

46222   46667 

47111 

497 


TABLE  XXXI.-USEFUL  NUMBERS  AND  FORMULJE. 


Title. 

Symbol. 

Number. 

Loga- 
rithm. 

Ratio  of  circumference  to  diameter  

Tt 

3.1415927 

0.4971409 

Reciprocal  of  same 

1 

0.3183099 

9.5028501 

Tt 

180° 

Degrees  in  arc  of  length  equal  to  radius  

Tt 

57.295780 

1.7581226 

10800' 

Minutes      "                        "          "          "     

3437.7468 

3.5362739 

Tt 

648000" 

Seconds     "                                              '     

206264.81 

5.3144251 

Tt 

Length  of  1°  arc  radius  unity  . 

Tt 

.01745329 

8.2418774 

Tt 

Length  of  1'  arc,      "           "    

.00029089 

6.4637261 

10800 

Lcnsrth  of  1"  arc        "           4t 

Tt 

.000004848 

4.6855749 

618000 

Radius  by  \vhich  1  foot  of  arc  =    1  degree  . 

57.295780 

1.7581226 

Radius  "        "       ^    "        "       =    1  minute. 

343.77468 

2.5362739 

Radius"        "      T£TT  "        "       =10  seconds 

206.28481 

2.3144251 

Factors  for  dividing  a  line  into  extreme  j 

0.6180340 

9.7910124 

and  mean  ratio  ) 

0..  3819660 

9.5820248 

Base  of  hyperbolic  logarithms  

0 

2.7182818 

0.4342945 

Modulus  of  common  system  of  logs.  =  log  £ 

M 

0.4342945 

9.6377843 

Reciprocal  of  same  =  hyp.  log.  10  

i 

2.3025851 

0.  "0221  57 

Length  of  seconds  pendulum  at  New  York 

in  inches  

39.11256 

1.51E81G2 

Length  of  seconds  pendulum  at  New  York 

ill  f  66  1 

3.25938 

0.5131350 

Acceleration  due  to  gravity  at  New  York.  .  . 

0 

32.1688 

1.5074347 

Square  root  of  same 

/~ 

5.67175 

0.7537173 

Yards  in  1  pietro 

U 

1.093623 

0.0388676 

Feet      in  1      "                                          

3.280869 

0.5159889 

Inches  in  1      u                          •        ..... 

39.37043 

1.5951701 

Metres  in  1  foot                                   

0.304797 

9.4840111 

Metres  in  1  yard                             

0.914392 

9.9611324 

Metres  in  1  mile                           .  .         

1609.330 

3.2066450 

498 


TABLE  XXXI. -USEFUL  NUMBERS  AND  FORMULAE. 


Title. 


Cubic  inches  in  1  U.  S.  gallon 

"  "      "    1  Imperial  gallon 

"      "    1  U.  S.  bushel 

Cubic  feet  in  1  U.  S.  gallon 

"         "    "  1  Imperial  gallon 

"    "  1  U.  S.  bushel 

Weight  of  1  cub.  foot  of  water,  barom.  30  in. 
ther.  39°.83  Fah. ;  pounds. . 

"    62° 
Weight  in  grains,  1  cubic  inch,  at  62°  Fah. . 

No.  of  grains  in  1  pound  avoir 

"        "        "  1  ounce      "    . 


Symbol.     Number. 


231. 
277.274 
2150.42 
0.133681 
0.160459 
1.244456 

62.379 
62.321 

252.458 
7000. 
437.5 


Loga- 
rithm. 


2.363G120 


3.3325233 
9.1260683 
9.2053655 
0.0949796 

1.7950384 
1.7946349 
2.4021892 
3.8450980 
2.6409781 


r   =  radius  of  circular  arc ; 

I    =  length  of  arc ; 

a°  =  degrees  in  same  arc. 


I     180° 

a°  =  — • . 

»•        it 

I     180° 
~  a0'     n 


I  =  a°r  . 


a' 


180° 


Radius  by  which  the  length  of  chord  c  in  feet  =  —  in  minutes; 


10  sin 


Hyp.  log  x  —  com.  log  x  X  -j-f  ,  or 


com.  log  (hyp.  log  x)  =  com.  log  (com.  log  x)  -f  0;3322157 
Com.  log  x  =  M  X  hyp.  log  x  ;  or 

com.  log  (com.  log  x)  =  9.6377843  -f  com.  log  (hyp.  log  x) 
Circumference  of  circle  (radius  =  r)  ................................ 

Area  of  circle  ....................................................... 

Area  of  sector  (length  of  arc  =  I)  .........................  .  .....  .... 

Area  of  sector  (angle  of  arc  =  a°)  ..................................  • 

Approximate  area  of  segment  (chord  =  c,  mid.  ord.  =  m)  ......... 

499 


—  - 


litr 


TTr2 


APPENDIX. 


Verification  of  eq.  (77). 


sin  8  0 

Eq.  (76)  p  = =-.  sin  Q  ,  cosec  — 


d,J  Q        i  00 

— Q-  =   cos  6  .  cosec  —  -  —  ,  sin  0  .  cot  —  .  cosec  —        (76J) 


d/j 

•        _.*•>  A     _  _  j_    ~    •  C7f7\ 


Verification  of  eq.  (81). 
Differentiating  eq.  (76$) 


d2p  0  2  00 

— G.r  =  -  sin  0  cosec  —   -    —  cos  0  cot   —  cosec  -^. 


l  i 

—2-  sin  0  cof   —    cosec --^   +  -^-3-  sin  0  coseca  -^ 


2P  0  p    {  0  0  \ 

=  ~  P  ~  ~N~  C0t  9     C0t    2\T  ^  "J\T»  \COt2    ]V   +  COSGC2    A^/ 

d-p  /  2  0          i  0 

"uo"j  =  p  \  ~  !  "  ^  cot  °  cot  ]y  +  A-*"  (a  cota  ,v 


APPENDIX.  501 


No\r 


dp  d2P 

in  which  substitute  for  —     -,  and  for  —   /-,  and  let 


1  ^5 

cot  Q  —    —  cot  -^   =  —  a 


4.  2/32  (_a)2  _  p2     _  i  _  1  cot  0  cot  |p  +  -~  (2  cot"  ^ 


a^-f          cotQ.cot        .--cof      .    - 


p 

2"  '         i  ~~i          o 

1-       +  a"-+       cot 


-|p  -fa  (a:-  I  cot    -) 


(1  4-  «2)- 

(81) 


10872 


Soi 


